Physical Chemistry 3: — Chemical Kinetics — F. Temps Institute of Physical Chemistry Christian-Albrechts-University Kiel Olshausenstr. 40 D-24098 Kiel, Germany Email:
[email protected] URL: http://www.uni-kiel.de/phc/temps Summer Term 2013
This scriptum contains notes for the lecture “Physical Chemistry 3: Chemical Kinetics” (MNF-chem0405) in the summer term 2013 at the Institute of Physical Chemistry of Christian-Albrechts-University (CAU) Kiel. This module is the last of the three semester lecture course in Physical Chemistry for B.Sc. students of Chemistry and Business Chemistry at CAU Kiel. It includes 3 hours weekly (3 SWS) for lectures and 1 hour weekly (1 SWS) for an exercise class.
c F. Temps (2004 — 2014) °
Last update: April 13, 2014
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Contents Preface
ix
Disclaimer
xi
Organisational matters
xii
1 Introduction
1
1.1 Types of chemical reactions
1
1.2 Time scales for chemical reactions
2
1.3 Historical events
5
1.4 References
7
2 Formal kinetics
8
2.1 Definitions and conventions
8
2.1.1 The rate of a chemical reaction
8
2.1.2 Rate laws and rate coefficients
9
2.1.3 The order of a reaction
10
2.1.4 The molecularity of a reaction
11
2.1.5 Mechanisms of complex reactions
12
2.2 Kinetics of irreversible first-order reactions
16
2.3 Kinetics of reversible first-order reactions (relaxation processes)
20
2.4 Kinetics of second-order reactions
24
2.4.1 A + A → products
24
2.4.2 A + B → products
25
2.4.3 Pseudo first-order reactions
26
2.4.4 Comparison of first-order and second-order reactions
27
2.5 Kinetics of third-order reactions
28
2.6 Kinetics of simple composite reactions
29
2.6.1 Consecutive first-order reactions
29
2.6.2 Reactions with pre-equilibrium
32
2.6.3 Parallel reactions
33
2.6.4 Simultaneous first- and second-order reactions*
33
2.6.5 Competing second-order reactions*
34
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2.7 Temperature dependence of rate coefficients
35
2.7.1 Arrhenius equation
35
2.7.2 Deviations from Arrhenius behavior
37
2.7.3 Examples for non-Arrhenius behavior
38
2.8 References 3 Kinetics of complex reaction systems
40 41
3.1 Determination of the order of a reaction
41
3.2 Application of the steady-state assumption
43
3.2.1 The reaction H2 + Br2 → 2 HBr
43
3.2.2 Further examples
45
3.3 Microscopic reversibility and detailed balancing
47
3.4 Generalized first-order kinetics*
48
3.4.1 Matrix method*
48
3.4.2 Laplace transforms*
53
3.5 Numerical integration
57
3.5.1 Taylor series expansions
58
3.5.2 Euler method
58
3.5.3 Runge-Kutta methods
59
3.5.4 Implicit methods (Gear):
61
3.5.5 Extrapolation methods (Bulirsch-Stoer)
62
3.5.6 Example: Consecutive first-order reactions
62
3.6 Oscillating reactions*
66
3.6.1 Autocatalysis
66
3.6.2 Chemical oscillations
67
3.7 References 4 Experimental methods
75 76
4.1 End product analysis
76
4.2 Modulation techniques
77
4.3 The discharge flow (DF) technique
78
4.4 Flash photolysis (FP)
82
4.5 Shock tube studies of high temperature kinetics
85
4.6 Stopped flow studies
86
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4.7 NMR spectroscopy
87
4.8 Relaxation methods
88
4.9 Femtosecond spectroscopy
89
4.10 References
93
5 Collision theory
94
5.1 Hard sphere collision theory
95
5.1.1 The gas kinetic collision frequency
95
5.1.2 Allowance for a threshold energy for reaction
99
5.1.3 Allowance for steric hindrance
100
5.2 Kinetic gas theory
101
5.2.1 The Maxwell-Boltzmann speed distributions of gas molecules in 1D and 3D
101
5.2.2 The rate of wall collisions in 3D
106
5.3 Transport processes in gases
110
5.3.1 The general transport equation
110
5.3.2 Diffusion
112
5.3.3 Heat conduction
113
5.3.4 Viscosity
114
5.4 Advanced collision theory
115
5.4.1 Fundamental equation for reactive scattering in crossed molecular beams
116
5.4.2 Crossed molecular beam experiments
117
5.4.3 From differential cross sections ( ) to reaction rate constants ( )
120
5.4.4 Laplace transforms in chemistry
121
5.4.5 Bimolecular rate constants from collision theory
123
5.4.6 Long-range intermolecular interactions
132
6 Potential energy surfaces for chemical reactions
138
6.1 Diatomic molecules
138
6.2 Polyatomic molecules
139
6.3 Trajectory Calculations
143
7 Transition state theory 7.1 Foundations of transition state theory 7.1.1 Basic assumptions of TST
146 146 147
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7.1.2 Formal kinetic model
149
7.1.3 The fundamental equation for ( )
149
7.1.4 Tolman’s interpretation of the Arrhenius activation energy
153
7.2 Applications of transition state theory
154
7.2.1 The molecular partition functions
154
7.2.2 Example 1: Reactions of two atoms A + B → A· · · B‡ → AB
156
7.2.3 Example 2: The reaction F + H2 → F· · · H· · · H‡ → FH + H
157
7.2.4 Example 3: Deviations from Arrhenius behavior (T dependence of ( ))*
159
7.3 Thermodynamic interpretation of transition state theory
160
7.3.1 Fundamental equation of thermodynamic transition state theory
160
7.3.2 Applications of thermodynamic transition state theory
164
7.3.3 Kinetic isotope effects
166
7.3.4 Gibbs free enthalpy correlations
168
7.3.5 Pressure dependence of
168
7.4 Transition state spectroscopy
170
7.5 References
174
8 Unimolecular reactions
175
8.1 Experimental observations
175
8.2 Lindemann mechanism
178
8.3 Generalized Lindemann-Hinshelwood mechanism
182
8.3.1 Master equation
182
8.3.2 Equilibrium state populations
184
8.3.3 The density of vibrational states ()
185
8.3.4 Unimolecular reaction rate constant in the low pressure regime
188
8.3.5 Unimolecular reaction rate constant in the high pressure regime
190
8.4 The specific unimolecular reaction rate constants ()
191
8.4.1 Rice-Ramsperger-Kassel (RRK) theory
191
8.4.2 Rice-Ramsperger-Kassel-Marcus (RRKM) theory
193
8.4.3 The SACM and VRRKM model
198
8.4.4 Experimental results
199
8.5 Collisional energy transfer*
200
8.6 Recombination reactions
201
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8.7 References 9 Explosions 9.1 Chain reactions and chain explosions
202 203 203
9.1.1 Chain reactions without branching
203
9.1.2 Chain reactions with branching
204
9.2 Heat explosions
207
9.2.1 Semenov theory for “stirred reactors” ( = const)
207
9.2.2 Frank—Kamenetskii theory for “unstirred reactors”
211
9.2.3 Explosion limits
212
9.2.4 Growth curves and catastrophy theory
212
10 Catalysis 10.1 Kinetics of enzyme catalyzed reactions (homogeneous catalysis)
214 214
10.1.1 The Michaelis-Menten mechanism
215
10.1.2 Inhibition of enzymes:
217
10.2 Kinetics of heterogeneous reactions (surface reactions)
218
10.2.1 Reaction steps in heterogeneous catalysis
218
10.2.2 Physisorption and chemisorption
218
10.2.3 The Langmuir adsorption isotherm
219
10.2.4 Surface Diffusion
222
10.2.5 Types of heterogeneous reactions
223
11 Reactions in solution
227
11.1 Qualitative model of liquid phase reactions
227
11.2 Diffusion-controlled reactions
230
11.2.1 Experimental observations
230
11.2.2 Derivation of
230
11.3 Activation controlled reactions
234
11.4 Electron transfer reactions (Marcus theory)
235
11.5 Reactions of ions in solutions
238
11.5.1 Effect of the ionic strength of the solution
238
11.5.2 Kinetic salt effect
238
11.6 References
240
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12 Photochemical reactions
241
12.1 Radiative processes in a two-level system; Einstein coefficients
241
12.2 Light amplification by stimulated emission of radiation (LASER)
242
12.3 Fluorescence quenching (Stern-Volmer equation)
243
12.4 The radiative lifetime 0 and the fluorescence quantum yield Φ
244
12.5 Radiationless processes in photoexcited molecules
245
12.5.1 The Born-Oppenheimer approximation and its breakdown
246
12.5.2 Avoided crossings of two potential curves of a diatomic molecule
249
12.5.3 Quantum mechanical treatment of a coupled 2×2 system
251
12.5.4 Radiationless processes in polyatomic molecules: Conical intersections
253
12.5.5 Femtosecond Time-Resolved Experiments
256
13 Atmospheric chemistry*
259
14 Combustion chemistry*
260
15 Astrochemistry*
261
16 Energy transfer processes*
262
17 Reaction dynamics*
263
18 Electrochemical kinetics*
264
A Useful physicochemical constants
265
B The Marquardt-Levenberg non-linear least-squares fitting algorithm
266
C Solution of inhomogeneous differential equations
270
C.1 General method
270
C.2 Application to two consecutive first-order reactions
271
C.3 Application to parallel and consecutive first-order reactions
273
D Matrix methods
276
D.1 Definition
276
D.2 Special matrices and matrix operations
277
D.3 Determinants
280
D.4 Coordinate transformations
281
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D.5 Systems of linear equations
283
D.6 Eigenvalue equations
285
E Laplace transforms
289
F The Gamma function Γ ()
291
G Ergebnisse der statistischen Thermodynamik
293
G.1 Boltzmann-Verteilung
294
G.2 Mittelwerte
296
G.3 Anwendung auf ein Zweiniveau-System
298
G.4 Mikrokanonische und makrokanonische Ensembles
301
G.5 Statistische Interpretation der thermodynamischen Zustandsgrößen
303
G.6 Chemisches Gleichgewicht
304
G.7 Zustandssumme für elektronische Zustände
306
G.8 Zustandssumme für die Schwingungsbewegung
307
G.9 Zustandssumme für die Rotationsbewegung
309
G.10 Zustandssumme für die Translationsbewegung
310
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Preface This scriptum contains lecture notes for the module “Physical Chemistry 3: Chemical Kinetics” (chem0405), the last part of the 3 semester course in Physical Chemistry for B.Sc. students of Chemistry and Business Chemistry at CAU Kiel. It is assumed (but not formally required) that students have previously taken courses on chemical thermodynamics (“Physical Chemistry 1: Chemical Equilibrium” or equivalent) and elementary quantum mechanics (“Physical Chemistry 2: Structure of Matter” or equivalent) as well as “Mathematics for Chemistry” (parts 1 and 2, or equivalent). Module chem0405 includes 3 hours weekly (3 SWS) for lectures and 1 SWS of exercises (56 contact hours combined). Students who started their studies at CAU Kiel should normally be in their 4th semester. The scriptum gives a summary of the material covered in the scheduled lectures to allow students to repeat the material more economically. It covers basic material that all chemistry students should learn irrespective of their possible inclination towards inorganic, organic or physical chemistry, but goes well beyond the standard Physical Chemistry textbooks used in the PC-1 and PC-2 courses. This is done in recognition of the established research focus at the Institute of Physical Chemistry at CAU to enable students to pursue their B.Sc. thesis project in Physical Chemistry. Some more specialized sections have been marked by asterisks and may be omitted on first reading towards the B.Sc. degree. Useful additional reference material is given in the Appendix. A brief lecture scriptum can never replace a textbook. For additional review, students are strongly encouraged (and at places required) to consult the recommended books on Chemical Kinetics beyond the standard Physical Chemistry textbooks. The book by Logan1 is a comprehensive overview of the field of chemical kinetics and a nice introduction; it also has the advantage that it is written in German. The book by Pilling and Seakins2 gives a good introduction especially to gas phase kinetics. The book by Houston3 includes more information on the dynamics of chemical reactions. The book by Barrante4 is highly recommended for everyone who wants to brush-up some math skills. A highly recommended web site for looking up mathematical definitions and recipies is the MathWorld online encyclopedia (http://mathworld.wolfram.com). The computer has become indispensable in modern research. This applies especially to Chemical Kinetics, where only computers can solve (by numerical integration) the coupled nonlinear partial differential equation systems that describe important complex reaction systems (e.g., related to combustion, the atmosphere or climate change). The integration of differential equations, a major task in chemical kinetics, can be performed using Wolfram Alpha (http://www.wolframalpha.com). More complicated problems can be solved with versatile computer algebra systems, e.g. (in order of increasing power) Derive,5 MuPad, MathCad,6 Maple, Mathematica7 . A graphics program (e.g., Origin,8
1 2 3 4 5 6 7 8
S. R. Logan, Grundlagen der Chemischen Kinetik, Wiley-VCH, Weinheim, 1996. M. J. Pilling, P. W. Seakins, Reaction Kinetics, Oxford University Press, Oxford, 1995. P. L. Houston, Chemical Kinetics and Reaction Dynamics, McGraw-Hill, 2001. J. R. Barrante, Applied Mathematics for Physical Chemistry, Prentice Hall, 2003 ($ 47.11). Derive is available as shareware without charge. MathCad is installed in the PC lab. Mathematica has been used by the author for some of the lecture material. Student versions of Origin are available for ≈ 35 − 120 (depending on run time).
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or QtiPlot) may be needed for data evaluation and presentation.9 As practically all original scientific literature is published in English (British and American; in fact many chemists at some time in their professional lifes have to master the subtle differences between both), this scriptum is written in English to introduce students to the predominating language of science at an early stage. Students should also note that the working language of some of the research groups (including that of the author) in the Chemistry Department at CAU is English. Finally, I would like to encourage all students to ask questions when they appear, whether during the lecture or in the exercise classes!
Kiel, April 13, 2014,
9
F. Temps
Excel also allows you to plot simple functions and data, but is not sufficient for serious scientific applications.
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Disclaimer Use of this text is entirely at the reader’s risk. Some sections are incomplete, some equations still have to be checked, some figures need to be redrawn, references are still missing, etc. No warranties in any form whatever are given by the author.
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Organisational matters • List of participants • Lecture schedule • Exams and grades — Homework assignments — Short questions — Written final exam (“Klausur”)
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Table 1: List of participants. Chemistry (B.Sc.) 82 Business Chemistry (B.Sc.) 25 Biochemistry (B.Sc.) Chemistry (2x-B.Sc.) Physics (M.Sc.) 1 Other) 1 Total 109 ) Not registered for exercise classes and final written exam (“Klausur”).
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Figure 1: Physical chemistry courses at CAU Kiel.
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Figure 2: Organisational matters.
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Figure 3: Requirements and grading.
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Figure 4: Lecture contents.
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Figure 5: Table of contents (continued).a
a
Topics marked with asterisks will be omitted in this course for lack of time. They are the subject of specialized lectures for advanced students.
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Figure 6: Recommended textbooks on chemical kinetics.
References to original publications and other recommended literature for further reading will be given at the end of each chapter.
1
1. Introduction 1.1 Types of chemical reactions • Homogeneous reactions: — Reactions in the gas phase, e.g., H2 + Cl2 → 2 HCl — Reactions in the liquid phase, e.g., H+ -transfer, reduction/oxidation reactions, hydrolysis, . . . (most of organic and inorganic chemistry) — Reactions in the solid phase (usually very slow, except for special systems and special conditions; not covered in this lecture). • Heterogeneous reactions: — Reactions at the gas-solid interface, e.g., C() + 12 O2 () → CO() — Reactions at the gas-liquid interface, e.g., heterogeneous catalysis, atmospheric reactions on aerosols (e.g., Cl release into the gas phase from polar stratospheric clouds in antarctic spring) ClONO2 () + HCl() → Cl2 () + HNO3 () — Reactions at the liquid-solid boundary, e.g., solvation, heterogeneous catalysis, electrochemical reactions (electrode kinetics), . . . • Irreversible reactions, e.g., H2 + 12 O2 → H2 O • Reversible reactions, e.g.,
H2 + I2 À 2 HI
• Composite reactions, e.g., CH4 + 2 O2 → CO2 + 2 H2 O • Elementary reactions, e.g., OH + H2 → H2 O + H CH3 NC → CH3 CN
1.2 Time scales for chemical reactions
2
1.2 Time scales for chemical reactions A chemical reaction requires some “contact” between the respective reaction partners. The “time between contacts” thus determines the upper limit for the rate of a reaction. I
Reactions in the gas phase: • Free mean pathlength À molecular dimension (diameter) . • Reaction rate is determined by time between collisions. • Time between two collisions @ :
y
≈ 500 m s
(1.1)
≈ 10−7 m
(1.2)
∆ ≈ ≈ 2 × 10−10 s
(1.3)
y • Gas phase reactions take place on the nanosecond time scale: I @ ≤
I
1 = 5 × 109 s−1 ∆
(1.4)
Reactions in the liquid phase: • Molecules are permanently in contact with each other. • Typical distance between neighboring molecules (upper limit allowing for solvation shell): (1.5) ≈ 2 × 10−8 10−7 cm • Maximal reaction rate is determined by the rate of diffusion of the reacting molecules in and out of the solvent cage.
I
Reactions in the solid phase: • Atoms and molecules localized on fixed lattice positions. • Reaction rate is determined by rate of diffusion (“hopping”) of the atoms and molecules via vacancies (unoccupied lattice positions, “Fehlstellen”) or interstitial sites (“Zwischengitterplätze”). • Hopping from one lattice position to another usually requires a corresponding high activation energy. • Solid state reactions are usually very slow.
1.2 Time scales for chemical reactions
I
3
Time scale of molecular vibrations: • The frequencies of the vibrational motions of the atoms in a molecule with respect to each other determine an upper limit for the rate of a unimolecular reaction. • Uncertainty principle (Fourier relation between frequency and time domain): ∆ ∆ ≥ ~
(1.6)
1 2
(1.7)
1 2 ∆˜
(1.8)
y ∆ ∆ ≥ With = = ˜: ∆ ≈
• Application to a C-C stretching vibration: y y
˜(C-C) ≈ 1000 cm−1
(1.9)
= = ˜ = 3 × 1013 s−1
(1.10)
= −1 = 333 × 10−14 s ≈ 33 fs
(1.11)
• Application to slow torsional motions of large molecular groups in solution: ˜ ≈ 100 cm−1
(1.12)
∆ ≈ 330 fs
(1.13)
y
I
Femtochemistry: We can observe ultrafast chemical reactions directly nowadays by following the motions of atoms or groups of atoms in the course of a reaction in real time by femtosecond spectroscopy and femtosecond diffraction techniques. This new area of “femtochemistry” has grown enormously in importance in the last decade, since the award of the Nobel prize in chemistry to A.H. Zewail in 1999. 1 femtosecond = 1 fs = 10−15 s
I
(1.14)
Timescale of e− -jump processes: Only electron jumps are faster than vibrational motions (by factor of roughly ). • Estimated time for − -jump to another orbital due to absorption/emission of a photon: ∆ ≈ 10−18 s (1.15)
There are indications that electron correlation may slow this time to some 10−17 s.10
10
The group of F. Krausz (MPI for Quantum Optics, Munich) recently used attosecond metrology based on an interferometric method to measure a delay of 21 ± 5 as in the photo-induced emission of electrons from the 2 orbitals of neon atoms with respect to emission from the 2 orbital by a 100 eV light pulse (Schultze et al, Science 328, 1658 (2010)).
1.2 Time scales for chemical reactions
4
• Other fast electron transfer processes: — Autoionization of a doubly excited state, by one electron falling to a lower orbital, while the other is excited into the ionization continuum. • But “chemical” intra- and intermolecular − -transfer reactions are again slower: — The electron jump itself is very fast (≈ 10−18 s, as above). — However, in the Franck-Condon limit, the reactions usually require substantial molecular rearrangements due to the structural differences between the reactants and products (⇒ Marcus theory, Section 11.4). The net reaction rate is thus again limited by the time scale of vibrational motion.
I
Conclusions: • The time scales of chemical reactions span the enormous range of some 30 orders of magnitude, from 1 fs = 10−15 s up to 109 y = 1016 s! • In view of this huge span, a factor-to-two experimental uncertainty in an experimental measurement or a theoretical prediction of a rate constant really means rather little. • The enormous range of reaction rates, their complex dependencies on temperature, pressure, and species concentrations, and the question of the underlying reaction mechanisms that determine the reaction rates, make the field of chemical reaction kinetics and dynamics very attractive and rewarding!
1.3 Historical events
5
1.3 Historical events I
Sucrose inversion reaction (Wilhelmy, 1850): Kinetic investigation of the H+ catalyzed hydrolysis of sucrose by monitoring the change as function of time of the optical activity of the solution from reactant (sucrose) to products (glucose and fructose).11
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Table 1.1: Optical activity parameters for the sucrose inversion reaction.
(1 M)) (1 M)) )
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C12 H22 O11 + H2 O −→ C6 H12 O6 + C6 H12 O6 sucrose glucose fructose ◦ ◦ +665 — +520 −920◦ ◦ Ø = +665 Ø = −200◦
(1 M) = specific angle of rotation of a 1 M solution at (Na-D).
Figure 1.1: Observed concentration-time profile of sucrose.
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This experiment is carried out in the Physical Chemistry Lab Course 1 at CAU (“Grundpraktikum”).
1.3 Historical events
I
6
Empirical rate law: ln
= − 0
(1.16)
Differentiation gives ln ln 0 − = − ( ) = − } | {z
(1.17)
=0
or
y Empirical rate law:
1 = −
(1.18)
= −
(1.19)
Reaction rate : ≡−
(1.20)
Notice that even though there is one molecule of H2 O taking part in the reaction, H2 O does not appear in the rate law, because it is practically constant in the dilute solution.
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Figure 1.2: Milestones in chemical kinetics.
1.3 Historical events
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1.4 References Arrhenius 1889 S. A. Arrhenius, Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren, Z. Physik. Chem. 4, 226 (1889). Bunker 1966 D. L. Bunker, Theory of Elementary Gas Reaction Rates, Pergamon, Oxford, 1966. Glasstone 1941 S. Glasstone, K. Laidler, H. Eyring, The Theory of Rate Processes, Mc Graw-Hill, New York, 1941. Homann 1975 K. H. Homann, Reaktionskinetik, Grundzüge der Physikalischen Chemie Bd. IV, R. Haase (Ed.), Steinkopf, Darmstadt, 1975. Houston 1996 P. L. Houston, Chemical Kinetics and Reaction Dynamics, McGrawHill, Boston, 2001. Johnston 1966 H. S. Johnston, Gas Phase Reaction Rate Theory, Ronald, New York, 1966. Kassel 1932 L. S. Kassel, The Kinetics of Homogeneous Gas Reactions, Chem. Catalog, New York, 1932. Logan 1996 S. R. Logan, Grundlagen der Chemischen Kinetik, Wiley-VCH, Weinheim, 1996. Pilling 1995 M. J. Pilling, P. W. Seakins, Reaction Kinetics, Oxford University Press, Oxford, 1995. Smith 1980 I. W. M. Smith, Kinetics and Dynamics of Elementary Gas Reactions, Butterworths, London, 1980. Steinfeld 1989 J. I. Steinfeld, J. S. Francisco, W. L. Haase, Chemical Kinetics and Dynamics, Prentice Hall, Englewood Cliffs, 1989.
1.4 References
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2. Formal kinetics 2.1 Definitions and conventions 2.1.1 The rate of a chemical reaction I
Rate of a simple reaction of type A + B → C + D: A B C D =− =+ =+ [B] [C] [D] [A] =− =+ =+ = −
≡ −
(2.1) (2.2)
We will usually write the concentrations of a species A by using square brackets, i.e., [A] and understand that [A] is time dependent, i.e., [A ()]. I
Definition of the rate of reaction:12 Using the standard convention for the signs of the stoichiometric coefficients of the reactants (−1) and products (+1), we write a generic chemical reaction as 1 B1 + 2 B2 + → B + +1 B+1 +
(2.3)
The progress of the reaction from reactants to products can be described using the reaction progress coordinate : = (2.4) The time dependence leads to the definition of the reaction rate. I
Definition 2.1: The reaction rate is defined as ≡
1
(2.5)
For the above reaction, therefore, =−
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1 2 1 1 +1 1 1 =− = = + =+ = | 1 | | 2 | | | | +1 |
(2.6)
Units:13 • SI unit for the reaction rate: [] = mol m3 s 12
13
(2.7)
Reaction rate = Reaktionsgeschwindigkeit. Von manchen Schulen wird auch der Begriff “Reaktionsrate” verwendet. Die bevorzugte “einzig wahre korrekte Übersetzung” wird von der jeweils anderen Seite sehr ernst genommen, allerdings gilt in Kiel in dieser Hinsicht Waffenstillstand. For general information on the SI system see (Mills 1988) or (Homann 1995).
2.1 Definitions and conventions
9
• Convenient units for gas phase reactions: — molar units: [] = mol cm3 s
(2.8)
[] = molecules cm3 s ⇒ cm−3 s−1
(2.9)
— molecular units:14
— We prefer to use molar units for rate constants, because rate constants are inherently quantities that are “averages” of some other quantity, e.g., a molecular cross section weighted by the relevant statistical distribution function (usually the Boltzmann distribution). Rate constants are thus properties of an ensemble of molecules, they are not original molecular quantities. • Convenient units for liquid phase reactions: [] = mol l s = M s
(2.10)
2.1.2 Rate laws and rate coefficients I
The rate equation (rate law): Experiments show that the reaction rate generally depends on the concentrations of the reactants. The functional dependence is expressed by the rate equation (also called rate law),15 e.g. 1 [B1 ] = | 1 | = ([B1 ] [B2 ] [B ] [B+1 ] )
= −
(2.11) (2.12)
for reaction 2.3. The rate laws of complex reactions (i.e., the ([B1 ] , [B2 ] , , [B ] , [B+1 ] , )) are often rather complicated; only rarely (as in the case of elementary reactions) are they simple and intuitive. Often (but not always), ([B1 ] , [B2 ] , , [B ] , [B+1 ] , ) can be written as a product of a rate coefficient and some power series of the concentrations, for example 1 [B1 ] = [B1 ] [B2 ] . . . =− (2.13) | 1 | Note that the exponents . . . are not usually equal to 1 2 . . . I
The rate constant (rate coefficient): The rate constant (also called rate coefficient)16 usually depends on temperature ( = ( )) and sometimes also on pressure ( = ( )), but not on the concentrations ( 6= ()). 14 15 16
IUPAC has decided that “molecules” is not a unit; see (Mills 1989) or (Homann 1995). Rate law = Geschwindigkeits- oder Zeitgesetz. Rate constant = Geschwindigkeitskonstante. Es wird manchmal auch die Übersetzung “Ratenkonstante” verwendet (aber bitte nicht “Ratekonstante”).
2.1 Definitions and conventions
I
10
Rate laws for first-order and second-order reactions: Simple rate laws describe first-order and second-order reactions: • Example of a first-order reaction: CH3 NC → CH3 CN =−
[CH3 NC] [CH3 CN] =+ = [CH3 NC]
(2.14)
• Example of a second-order reaction: C4 H9 + C4 H9 → C8 H18 =−
1 [C4 H9 ] [C8 H18 ] =+ = [C4 H9 ]2 2
(2.15)
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Question: How can we determine reaction rates and rate constants ?
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Figure 2.1: The easiest — and always applicable — method for determining the reaction rate and rate coefficient is the tangent method. To determine , you simply take the slope of a plot of () at time ; then follows via Eq. 2.13.
We’ll learn about better methods for determining for elementary reactions (first-order and second-order reactions) further below.
2.1.3 The order of a reaction I
Definition 2.2: The order of a reaction is defined as the sum of the exponents in the rate law (Eq. 2.13).
2.1 Definitions and conventions
I
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Example: Considering a generic rate law, like −
[A] = | | [A] [B]
(2.16)
the total order of the reaction is = + . We also say, the reaction is -th order in [A] and -th order in B, . . . The order of a reaction may be determined by the tangent method (Fig. 2.1): Its effects on the reaction rate are observable by increasing (e.g., doubling) the concentrations of the reaction partners one by one. Better methods will be explored later (⇒ homework assignments). I
Caveats: • The order of a reaction is, in general, an empirical quantity; it has to be determined by an experimental measurement. • The order of a complex reaction cannot be determined by simply looking at the overall reaction. • There are reactions with integer order, like the first-order or second-order reactions above (Eqs. 2.14 and 2.15). However, the order can also be fractional, as for the following reactions: (1) CH3 CHO → CH4 + CO: [CH4 ] = [CH3 CHO]32
(2.17)
[HBr] 0 [H2 ] [Br2 ]12 = 1 + 00 [HBr] [Br2 ]
(2.18)
(2) H2 + Br2 → 2 HBr:
• A negative order in a concentration means that the respective species acts as an inhibitor. • A positive order in a product concentration means that the reaction rate is enhanced by autocatalysis.
2.1.4 The molecularity of a reaction I
Definition 2.3: The molecularity of a reaction is the number of molecules in an elementary reaction step at the microscopic (molecular) level.
2.1 Definitions and conventions
I
12
There are only three possible cases for the molecularity (one example for each case): • monomolecular (= unimolecular) reactions:
• bimolecular reactions:
CH3 NC → CH3 CN
(2.19)
O + H2 → OH + H
(2.20)
H + O2 + M → HO2 + M
(2.21)
• termolecular reactions:
• Events where more than three species come together in the right configuration and react are extremely unlikely and do not play a role.
I
Caveat: The order of a reaction can be integer or non-integer. However, the molecularity can be only 1 or 2, or at the most 3.
2.1.5 Mechanisms of complex reactions A stoichiometric formula rarely describes the reactive events happening at the microscopic, molecular level. It is, for example, completely impossible that the combustion of 1 -heptane molecule with 11 molecules of O2 -C7 H16 + 11 O2 → 7 CO2 + 8 H2 O
(2.22)
in an internal combustion engine occurs in a single collision and gives 15 product molecules. Instead, the net reaction is the result of a complex sequence of a very large number (thousands, in this case) of unimolecular, bimolecular, and termolecular elementary reactions. The identification of the individual elementary reactions, the determination of their rates as a function of and , and the verification of the ensuing reaction mechanism is one of the main tasks in the field of chemical kinetics. I
Examples for complex reaction systems: • H2 /O2 system (Fig. 2.2), • Combustion of CH4 (Fig. 2.3), • NO formation and removal in flames (Fig. 2.4).
2.1 Definitions and conventions
13
I
Figure 2.2: Reactions in the H2 /O2 system.
I
Rate equations for the H2 /O2 system: Assuming that we know the reaction mechanism (i.e., all elementary reactions contributing), we can write down the rate equations for all species. Doing so, we obtain a set of coupled non-linear differential equations (DE’s) which can be solved usually only numerically, if we want to describe the net reaction: [H] = −1 [H] [O2 ] + 2 [O] [H2 ] + 3 [OH] [H2 ] − 4 [H] [O2 ] [M] −5 [H] [HO2 ] − 6 [H] [HO2 ] + [OH] = 1 [H] [O2 ] + 2 [O] [H2 ] − 3 [OH] [H2 ] + 26 [H] [HO2 ] −7 [OH] [HO2 ] − 28 [OH]2 [M] + [O] =
(2.23)
(2.24) (2.25) (2.26)
2.1 Definitions and conventions
I
Figure 2.3: Mechanism of CH4 combustion (Warnatz 1996).
I
Figure 2.4: NO in combustion processes (Warnatz 1996).
14
2.1 Definitions and conventions
I
15
How we will proceed: • In the remainder of this Chapter, we shall learn how to analytically solve the rate equations for elementary reactions and for simple composite reactions. • In Chapter 3, we shall then consider different methods for solving coupled nonlinear differential equations for complex reaction systems. — These methods are well established for stationary reaction systems. — However, the modeling of instationary reaction systems (such as ignition processes, explosions and detonations, 3D or 4D models of atmospheric chemistry and climate evolution, . . . ) continues to be a tremendously challenging task.
2.2 Kinetics of irreversible first-order reactions
16
2.2 Kinetics of irreversible first-order reactions
A→B I
(2.27)
Solution of the rate equations for A and B: − Mass balance:
[A] [B] =+ = [A] [A] [B] + =0
(2.28)
(2.29)
Integration of the rate equation (Eq. 2.28): [A] = [A] Z Z Z ln [A] = − = − −
y
(2.30) (2.31)
ln [A] = − +
(2.32)
[A ( = 0)] = [A]0
(2.33)
= ln [A]0
(2.34)
ln [A] = − + ln [A]0
(2.35)
[A ()] = [A]0 −
(2.36)
[B ()] = [A]0 − [A ()]
(2.37)
Initial value condition at = 0:
y Solution for [A ()]: y
Solution for [B ()]: y
¡ ¢ [B ()] = [A]0 1 − −
(2.38)
2.2 Kinetics of irreversible first-order reactions
17
I
Figure 2.5: Kinetics of first-order reactions.
I
Time constant: The rate coefficient has the dimension of an inverse time. Thus, =
1
(2.39)
1 has dimension of time and is the time after which [A] has dropped to (= 368 %) of its initial value: [A]0 [A]0 ≈ (2.40) [A]= = [A]0 − = 2718 I
Half lifetime: The half lifetime 12 is the time after which [A] has dropped to its initial value (Fig. 2.5):
y
£ ¡ ¢¤ [A]0 A = 12 = [A]0 − 12 = 2 1 2 = ln 2
1 of 2
(2.41)
− 12 =
(2.42)
12
(2.43)
y 12 =
ln 2
(2.44)
2.2 Kinetics of irreversible first-order reactions
I
18
Conclusion: The half lifetime for a first-order reactions is independent of the initial concentration!
I
Experimental determination of k: (1) Plot of ln [A] vs. gives a straight line with slope = − (see Fig. 2.5): ln ([A] [A]0 ) = −
(2.45)
Advantage: We do not need to know the absolute concentration of [A]. (2) Numerical fit of to the exponential decay curve using the Marquardt-Levenberg non-linear least-squares fitting algorithm (Fig. 2.6; see Appendix B for details). This method is useful in particular for analyzing experimental decay curves with a constant background term. Again, we do not need to know the absolute concentration of [A]. Advantages: Can be applied in presence of a constant background signal (noise, weak DC signal). Can also be applied for analyzing data from pump-probe experiments, when the duration of the pump is not short to the decay (deconvolution).
I
Figure 2.6: Exponential decay curve of a particular vibration-rotation state of the CH3 O radical resulting from its unimolecular dissociation reaction according to CH3 O → H + H2 CO (Dertinger 1995). The small box is the output box from a fit using the software package ORIGIN.
2.2 Kinetics of irreversible first-order reactions
19
I
Figure 2.7: Excited-state relaxation dynamics of the adenine DNA dinucleotide after UV excitation.
I
Note (and warning): Anybody who wants to do least-squares fitting is urged to consult appropriate literature before doing the fitting! The book by Bevington17 is considered required reading for any physical chemistry graduate student. As the saying goes: With three parameters, you can draw an elephant. With a fourth parameter, you can make him walk.
17
P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, Boston, 1992.
2.3 Kinetics of reversible first-order reactions (relaxation processes)
20
2.3 Kinetics of reversible first-order reactions (relaxation processes) 1
A À B
(2.46)
[A] = −1 [A] + −1 [B]
(2.47)
[A] [B] =− = −1 [A] + −1 [B] = 0
(2.48)
[B]∞ 1 = = [A]∞ −1
(2.49)
−1
I
I
Rate equation:
Equilibrium at t → ∞: y
• The rate constant for the reverse reaction −1 can be calculated from 1 and . • Or the equilibrium constant can be determined from measurements of the forward and reverse rate constants. From , we can then determine important thermochemical quantities, for example for reactive free radicals, via ∙ ¸ ∆ ª (B ª ) (C ª ) = exp − (2.50) = (A ª ) • Important note: — In the frequently encountered case that the number of species on the reactant and product side of a reaction differ, one has to be very careful with the units of and . Consider, for example, a gas phase reaction of the type A→B+C
In this case, has the dimension of mol cm3 : B C 0 1 B C 1 [B] [C] = = = A = = −1 [A] A 0 has the dimension of bar: B C 0 = A But the thermodynamic equilibrium constant is dimensionless: ¸ ∙ 0 (B ª ) (C ª ) ∆ ª = ª = exp − = (A ª ) — General relation between 0 , and : ¡ ¢−Σ ( ) ¡ ¢Σ ( ) = × ª = 0 × ª
(2.51)
(2.52)
(2.53)
(2.54)
(2.55)
2.3 Kinetics of reversible first-order reactions (relaxation processes)
I
21
Solution of the rate equations for [A] and [B]: Returning to the time dependence of the reaction 1
A À B
(2.56)
[A] = −1 [A] + −1 [B]
(2.57)
−1
we now solve the rate equation
• Mass balance:
[B] = [A]0 − [A]
(2.58)
• Integration: [A] = −1 [A] + −1 [B] = −1 [A] + −1 ([A]0 − [A]) = − (1 + −1 ) [A] + −1 [A]0 y
[A] = − (1 + −1 ) [A] − −1 [A]0
(2.59) (2.60) (2.61)
(2.62)
• Solution by substitution: = (1 + −1 ) [A] − −1 [A]0
(2.63)
= (1 + −1 ) [A]
(2.64)
(1 + −1 )
(2.65)
[A] = y
= − (1 + −1 )
(2.66)
= − (1 + −1 )
(2.67)
ln = − (1 + −1 ) +
(2.68)
y • Initial value condition at = 0:
= 0
(2.69)
= ln 0
(2.70)
= exp [− (1 + −1 ) ] 0
(2.71)
y y
2.3 Kinetics of reversible first-order reactions (relaxation processes)
I
22
Solution for [A (t)] (Fig. 2.8):
or
(1 + −1 ) [A] − −1 [A]0 = exp [− (1 + −1 ) ] (1 + −1 ) [A]0 − −1 [A]0
(2.72)
−1 [A]0 1 + −1 = exp [− (1 + −1 ) ] −1 [A]0 − [A]0 1 + −1
(2.73)
[A] −
This expression is not so easy to memorize, but we may recast it in a simple way: Using
which gives
[B]∞ [A]0 − [A]∞ 1 = = −1 [A]∞ [A]∞ [A]∞ =
we obtain
−1 [A]0 1 + −1
[A ()] − [A]∞ = exp [− (1 + −1 ) ] [A]0 − [A]∞
(2.74)
(2.75)
(2.76)
With ∆ [A] = [A ()] − [A]∞ and ∆ [A]0 = [A]0 − [A]∞ , we write this result in compact form as ∆ [A] = exp [− (1 + −1 ) ] (2.77) ∆ [A]0
I
Figure 2.8: Kinetics of reversible first-order reactions.
2.3 Kinetics of reversible first-order reactions (relaxation processes)
I
23
Relaxation processes: • The transition from a deviation from equilibrium into the equilibrium is called relaxation. • (1 + −1 )−1 has the dimension of time and is called the relaxation time, = (1 + −1 )−1
(2.78)
• The temporal evolution of relaxation processes from a nonequilibrium to the equilibtium state is determined by the sum of the respective forward and reverse rate constants!
2.4 Kinetics of second-order reactions
24
2.4 Kinetics of second-order reactions 2.4.1 A + A → products
I
A+A→P
(2.79)
1 [A] = [A]2 2
(2.80)
Solution of the rate equation: − Integration: [A] = −2 [A]2 y
(2.81)
1 = −2 + [A]
(2.82)
[A ( = 0)] = [A]0
(2.83)
− Initial value condition at = 0:
y =−
1 [A]0
(2.84)
Solution for [A ()] (Fig. 2.15): 1 1 = + 2 [A] [A]0
I
(2.85)
Experimental determination of k: (1) Graphical method: 1 1 = + 2 [A] [A]0 ⇒ Plot of [A]−1 vs. gives a straight line with slope = 2 (Fig. 2.9). ⇒ We do need to know the absolute concentration of [A]!
(2) We can again use a nonlinear least squares fitting method (Appendix B).
(2.86)
2.4 Kinetics of second-order reactions
25
I
Figure 2.9: Kinetics of second-order reactions.
I
Half lifetime: y
£ ¡ ¢¤ [A]0 A 12 = 2
2 1 = + 2 12 [A]0 [A]0
y 12 =
1 2 [A]0
(2.87) (2.88)
(2.89)
y The half lifetime for a second-order reactions depends on the initial concentration!
2.4.2 A + B → products A+B→P
(2.90)
2.4 Kinetics of second-order reactions
I
26
Solution of the rate equation: −
[B] [A] =− = [A] [B]
(2.91)
(1) Substitution:
y
= ([A]0 − [A] ) = ([B]0 − [B] )
(2.92)
[A] = [A]0 −
(2.93)
= ([A]0 − ) ([B]0 − )
(2.94)
(2) Integration using partial fractions (skipped here): (3) General solution: [B]0 [A] = exp ([A]0 − [B]0 ) [A]0 [B]
(2.95)
2.4.3 Pseudo first-order reactions In experimental studies of second-order reactions of the type A + B → C, one likes to use a high excess of one of the reactions partners (e.g., [B] À [A]) so that [B] ≈ const Under this condition, the reaction is said to be pseudo first-order, because [B] = 0 y Solution of [A ()]:
I
0
[A] = [A]0 − = [A]0 −[B]
(2.96) (2.97)
Experimental investigation of pseudo first-order reactions: (1) Measurements of the pseudo first-order decay of [A] vs. at different concentrations of [B]: Plot of ln [A] vs. give pseudo first-order rate constant 0 = [B] for each value of [B]. (2) Plot of 0 = [B] vs. [B] gives straight line with slope . ⇒ We do not need the absolute concentration of [A], only the concentration of [B] is needed.
2.4 Kinetics of second-order reactions
I
27
Figure 2.10: Kinetics of pseudo first-order reactions.
2.4.4 Comparison of first-order and second-order reactions Consider a first-order and a second-order reaction with the same initial concentrations and the same initial rate at = 0. As shown in Fig. 2.11 below, the rate of the second-order reaction rapidly slows down as time increases. I
Figure 2.11: Comparison of first-order and second-order reactions.
2.5 Kinetics of third-order reactions
28
2.5 Kinetics of third-order reactions
y − I
A+B+C→D
(2.98)
[A] = [A] [B] [C]
(2.99)
Pseudo second-order combination reactions: In combination reactions of two species A + A → A2 or A + B → AB (for example, the recombination of two free radicals), one of the species (bath gas M) is usually present in large excess ([M] À [A], [B]), so that the kinetics becomes pseudo second-order, giving the two cases: • •
A + A + M → A2 + M
(2.100)
A + B + M → AB + M
(2.101)
M plays the role of an inert collision partner which removes the excess energy from the newly formed A2 or AB molecules.
2.6 Kinetics of simple composite reactions
29
2.6 Kinetics of simple composite reactions 2.6.1 Consecutive first-order reactions
1
(2.102)
2
(2.103)
A −→ B
B −→ C I
Coupled differential equations: [A] = −1 [A] [B] = +1 [A] − 2 [B] [C] = +2 [B]
(2.104) (2.105) (2.106)
In this case, since the DE’s are linear, there is an exact solution which we will examine first. We will also look at an approximate solution which can be obtained with the quasi steady-state approximation and at methods to obtain numerical solutions, which we have to use for complex non-linear inhomogeneous DE systems. I
Mass balance:
I
Initial conditions:
I
[A] + [B] + [C] = [A]0
(2.107)
[A ( = 0)] = [A]0 [B ( = 0)] = 0 [C ( = 0)] = 0
(2.108) (2.109) (2.110)
Exact solution: • First-order decay of [A]: y
[A] = −1 [A]
(2.111)
[A] = [A]0 −1
(2.112)
[B] = +1 [A] − 2 [B]
(2.113)
• Inhomogeneous DE for [B]:
2.6 Kinetics of simple composite reactions
30
• Solution for [B ()]18 :
⎧ ¢ 1 [A]0 ¡ −1 ⎪ ⎨ − −2 if 1 = 6 2 [B ()] = 2 − 1 ⎪ ⎩ 1 [A]0 −1 if 1 = 2
(2.114)
• Reaction products by mass balance:
[C ()] = [A]0 − [A ()] − [B ()] =
I
Figure 2.12: Kinetics of consecutive first-order reactions (case I).
18
The solution is detailed in Appendix C.
(2.115)
2.6 Kinetics of simple composite reactions
31
I
Figure 2.13: Kinetics of consecutive first-order reactions (case II).
I
Caveat: As seen, the general solution for [B] is the difference of two exponentials, [B] =
¢ 1 [A]0 ¡ −1 − −2 , 2 − 1
(2.116)
One exponential describes the rise, the other the fall of [B]. It is sometimes implicitly, but wrongly assumed that the rise time corresponds to 1 and the decay time to 2 . The truth is that we cannot tell just from the shape of the concentration-time profile whether 1 or 2 correspond to the rise or to the fall of [B]. I
Quasi steady-state approximation: Under the condition that 2 À 1
(2.117)
we can find a simple solution for the DE’s. Under this condition, after a short initial induction time (see Fig. 2.13), the change of [B] is very small compared to that of [A]. Therefore, we have approximately [B] ≈0
(2.118)
This important approximation is known as the (quasi)steady-state approximation.19
19
Deutsch: Quasistationaritätsannahme.
2.6 Kinetics of simple composite reactions
I
32
Quasi steady-state solution for [B (t)]: For we have [B] = 1 [A] − 2 [B] ≈ 0
(2.119)
1 [A] ≈ 2 [B]
(2.120)
y y [B] =
1 1 [A] = [A]0 −1 2 2
(2.121)
For , [C ()] is therefore given by [C] = 2 [B] ≈ 1 [A] y
¡ ¢ [C] = [A]0 1 − −1
(2.122) (2.123)
2.6.2 Reactions with pre-equilibrium
1
A À B
(2.124)
B → C
2
(2.125)
1 −1 À 2
(2.126)
1 [B] = = [A] −1
(2.127)
[B] = [A]
(2.128)
[C] = 2 [B] = 2 [A]
(2.129)
−1
I
Equilibrium between [A] and [B]:
y
I
Time dependence of [C]:
2.6 Kinetics of simple composite reactions
33
2.6.3 Parallel reactions
A −→ B
1
(2.130)
2
(2.131)
3
(2.132)
A −→ C
A −→ D y y
[A] = − (1 + 2 + 3 ) [A]
(2.133)
[A] = [A]0 −(1 +2 +3 )
(2.134)
and ¡ ¢ 1 [A]0 1 − −(1 +2 +3 ) (1 + 2 + 3 ) ¡ ¢ 2 [A]0 1 − −(1 +2 +3 ) [C] = (1 + 2 + 3 ) ¡ ¢ 3 [A]0 1 − −(1 +2 +3 ) [D] = (1 + 2 + 3 ) [B] =
(2.135) (2.136) (2.137)
Note that the decay of [A] is determined by the total removal rate constant = 1 + 2 + 3
2.6.4 Simultaneous first- and second-order reactions*
1
(2.138)
2
(2.139)
A −→ C
A + A −→ D y y
I
[A] = −1 [A] − 22 [A]2 ¶ µ 1 22 1 22 exp (−1 ) + + =− [A] 1 1 [A]0
(2.140) (2.141)
Approximation for k1 t ¿ 1: Power series expansion of the exponential gives ¶ µ 1 1 1 (2.142) ≈ + + 22 × [A] [A]0 [A]0
2.6 Kinetics of simple composite reactions
34
2.6.5 Competing second-order reactions*
1
(2.143)
2
(2.144)
3
(2.145)
A + A −→ A2
A + B −→ P1
B −→ P2
I
Solution for the case [A] À [B]: (1) [A] = −21 [A]2 − 2 [A] [B] ≈ −21 [A]2 y
1 1 − = 21 [A] [A]0
(2.146) (2.147) (2.148)
or ¢−1 ¡ [A] = [A]0 −1 + 21 [A]0 = 1 + 21 [A]0
(2.149) (2.150)
(2) [B] = −2 [A] [B] − 3 [B] (2.151) This is another inhomogeneous first-order DE, for which the solution can be found as outlined in Appendix C, giving [B] = [B]0 × (1 + 21 [A]0 )−2 21 × exp (−3 ) I
(2.152)
Example: 1
(2.153)
2
(2.154)
3
(2.155)
CH3 + CH3 −→ C2 H6
CH3 + OH −→ CH2 + H2 O OH −→ P2
In the experimental study of reaction 2.154 (Deters 1998), the rate coefficient for the reaction (2 according to Eq. 2.152) was fitted to measured OH concentration-time profiles in the presence of high concentrations of CH3 using the Marquardt-Levenberg nonlinear least squares fitting algorithm (Press 1992). The rate coefficients 1 and 3 were determined by independent measurements. The absolute CH3 concentrations, which are needed for the evaluation of 2 according to Eq. 2.152, were obtained using a quantitative titration reaction.
2.7 Temperature dependence of rate coefficients
35
2.7 Temperature dependence of rate coefficients 2.7.1 Arrhenius equation I
Arrhenius:20 Svante Arrhenius (Arrhenius 1889) found the following empirical relation describing the temperature dependence of the reaction rate constant: ln =− (1 )
I
Definition 2.4: is called the Arrhenius activation energy of the reaction: = −
I
(2.156)
ln (1 )
(2.157)
It is very important to keep in mind the following points: • Eq. 2.156 is nothing else but a simple and very convenient way to describe the empirical temperature dependence of reaction rate constants! • Eq. 2.156 says that, within experimental error, a plot of ln vs. 1 should give a straight line. We will see later, however, that this is rarely exactly true. The study and the understanding of non-Arrhenius behavior is in fact an important topic in modern kinetics. • as determined by Eqs. 2.156 and 2.157 is therefore a purely empirical quantity! • We will use Eq. 2.157 simply as the definition and recipy for determining from experimental data and to relate theoretical models to the experimental data!
I
The rationale behind the Arrhenius equation (2.156): The rationale for the Arrhenius equation is that molecules need additional energy so that chemical bonds can be broken, atoms can rearrange, and new bonds can be formed (Fig. 2.14). Van’t Hoff’s equation: − ← → − ∆ = = → − ← − 2 ln 2 ln = − → − ← − = − 2 ln
Thus we can write
20
ln = 2
2 ln
(2.158) (2.159) (2.160)
(2.161)
Arrhenius is generally regarded as one of the founders of physical chemistry (together with Faraday, van’t Hoff, and Ostwald).
2.7 Temperature dependence of rate coefficients
or21
ln =− (1 )
36
(2.164)
I
Figure 2.14: Derivation of the Arrhenius equation.
I
Experimental determination of E (see Fig. 2.14): A plot of ln vs. 1 gives a straight line with slope −
I
Integration of Eq. 2.156: Assuming that is independent of , Eq. 2.156 can be integrated: (2.165) ln = 2 y + (2.166) ln = − y ( ) = − (2.167) Equation 2.167 is used to represent ( ) over a limited -range using the two parameters and : • = Arrhenius activation energy (measure for the energy barrier of the reaction) • = pre-exponential factor (frequency factor, collision frequency). 21
(1 ) 1 =− 2
(2.162)
= − 2 (1 )
(2.163)
y
2.7 Temperature dependence of rate coefficients
I
37
It is important to realize, however, that the value of E is not equal to the true height of the potential energy barrier for the reaction! As said above, is an empirical quantity! Its relation to the true threshold energy 0 , which is the minimum energy above which reaction may occur, will be a main subject in the theory of bimolecular and unimolecular reactions (Sections 5 — 8). There is also a statistical interpretation of (see Section 8).
I
Figure 2.15: The reaction HCO → H + CO (G. Friedrichs).
2.7.2 Deviations from Arrhenius behavior In general, is dependent on temperature, i.e., = ( ). For gas phase reactions, √ for example, there is at least the dependence of the hard sphere collision frequency. The -dependence of is often small compared to so that within experimental error the dependence of ( ) is determined by . If the -dependence of is not small compared to , we will have to use a more general expression for ( ), as explained in the following. I
Generalized three-parameter expression for k (T ): To account for deviations from Arrhenius because of ( ), one conveniently employs the three parameter expression ( ) = −0 (2.168) with and 0 now being -independent quantities. Range of values for specific bimolecular reactions (justified by theory; see later): − 15 ≤ ≤ +3
(2.169)
2.7 Temperature dependence of rate coefficients
38
I
Question: What is the relation of Eq. 2.168 with the Arrhenius equation?
I
Answer: We have to evaluate the Arrhenius parameters from Eq. 2.168 according to the definitions of and : (1) : ln µ ¶ 0 2 ln + ln − = +
= + 2
(2.170) (2.171)
y = 0 +
(2.172)
Only in the case that ¿ 0 can we neglect the term compared to 0 ! (2) : From the expression for , we have 0 = −
(2.173)
( ) = −( − ) = − = −
(2.174) (2.175) (2.176)
y
Comparing coefficients, we find =
(2.177)
2.7.3 Examples for non-Arrhenius behavior Complex forming bimolecular reactions may show extreme non-Arrhenius behavior. An extreme case is the reaction OH + CO → H + CO2 , which incidentally is the single most important reaction in hydrocarbon combustion next to H + O2 → OH + O.
2.7 Temperature dependence of rate coefficients
I
39
Figure 2.16: The reaction OH + CO → H + CO2 (Fulle 1996, Kudla 1992).
Another case is encountered if tunneling plays a role (for example, H+ transfer reactions at low temperatures). In this case, ( ) tends to a plateau value as → 0 (Eisenberger 1991).
2.7 Temperature dependence of rate coefficients
40
2.8 References Arrhenius 1889 S. A. Arrhenius, Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren, Z. Physik. Chem. 4, 226 (1889). Bevington 1992 P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, Boston, 1992. Dertinger 1995 S. Dertinger, A. Geers, J. Kappert, F. Temps, J. W. Wiebrecht, Rotation-Vibration State Resolved Unimolecular Dynamics of Highly Vibrationally Excited CH3 O (2 ): III. State Specific Dissociation Rates from Spectroscopic Line Profiles and Time Resolved Measurements, Faraday Discuss. Roy. Soc. 102, 31 (1995). Deters 1998 R. Deters, M. Otting, H. Gg. Wagner, F. Temps, B. László, S. Dóbé, T. Bérces, A Direct Investigation of the Reaction CH3 + OH at = 298 K: Overall Rate Coefficient and CH2 Formation, Ber. Bunsenges. Phys. Chem. 102, 58 (1998). Eisenberger 1991 H. Eisenberger, B. Nickel, A. A. Ruth, W. Al-Soufi, K. H. Grellmann, M. Novo, Keto-Enol Tautomerization of 2-(20 -Hydroxyphenyl)benzoxazole and 2-(20 -Hydroxy-40 -methylphenyl)benzoxazole in the Triplet State: Hydrogen Tunneling and Isotope Effects. 2. Dual Phosphorescence Studies, J. Phys. Chem. 95, 10509 (1991). Fulle 1996 D. Fulle, H. F. Hamann, H. Hippler, J. Troe, High Pressure Range of Addition Reactions of HO. II. Temperature and Pressure Dependence of the Reaction HO + CO À HOCO → H + CO2 , J. Chem. Phys. 105, 983 (1996). Homann 1995 K. H. Homann, SI-Einheiten, VCH, Weinheim, 1995. Kudla 1992 K. Kudla, A. G. Koures, L. B. Harding, G. C. Schatz, A quasiclassical trajectory study of OH rotational excitation in OH + CO collisions using ab initio potential surfaces, J. Chem. Phys. 96, 7465 (1992). Mills 1988 I. M. Mills et al., Units and Symbols in Physical Chemistry, Blackwell, Oxford, 1988. Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1992. Versions are also available for C and Pascal. Warnatz 1996 J. Warnatz, U. Maas, R. W. Dibble, Combustion. Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation, Springer, Heidelberg, 1996.
2.8 References
41
3. Kinetics of complex reaction systems Real reaction systems are usually much more complicated: • Net reaction mechanisms may consist of ≈ 100 - 10 000 elementary reactions. • No analytical solutions for the rate laws. • In favorable cases, one may be able to use certain approximations to simplify the reaction systems, like — apply steady-state approximation where possible, — take into account reactions in equilibrium, — take into account microscopic reversibility (detailed balancing). • For more acurate descriptions, however, one usually needs numerical solutions for the rate equations. Using modern computers, this is not a problem even for 10 000 reactions, as long as the transport processes are not too complicated (as, for example, 1D reaction systems such premixed flames or 1D models of atmospheric chemistry). However, numerical solutions are still extremely challenging for instationary 3D reaction systems, such as — 3D models of atmospheric chemistry including daily/annual variations and couplings to the ocean, — ignition processes (internal combustion engines).
3.1 Determination of the order of a reaction Any reaction mechanism which we may postulate has to explain the observed reaction order. Thus, the determination of the reaction order is a central topic. We explore some general methods for determining the reaction order by considering an example: [A] =− (3.1) = [A] I
First-order and second-order reactions: Test whether plots of ln vs. or 1 vs. , give the straight lines expected for first-order or second-order reactions, respectively.
I
Log-log plot of reaction rate vs. concentration: We see from Eq. 3.1 that a log-log plot of vs. [A] gives a straight line with slope : lg ∝ lg [A]
(3.2)
3.1 Determination of the order of a reaction
I
42
Half lifetime method (for n 6= 1):22 − y
Z
y −
[A]
[A] = [A]
−
[A] =
Z
−
1 [A]−(−1) = − + ( − 1)
(3.3)
(3.4) (3.5)
Initial value condition at = 0: [A ( = 0)] = [A]0 y =− y
Half lifetime: y
1 1 = ( − 1) −1 − [A] [A]−1 0 £ ¡ ¢¤ 1 A 12 = [A]0 2
2−1 1 = ( − 1) 12 −1 − [A]0 [A]−1 0
y 12 = y
1 [A]0−(−1) ( − 1)
2−1 − 1 1 ( − 1) [A]−1 0
lg 12 ∝ −( − 1) lg [A]0
(3.6)
(3.7)
(3.8)
(3.9) (3.10)
(3.11) (3.12)
y A log-log plot of 12 vs. [A]0 gives a straight line with slope −( − 1).
22
We leave aside a usually occuring stoichiometric factor ν here, since we can always define νk = k0 as rate constant k0 .
3.2 Application of the steady-state assumption
43
3.2 Application of the steady-state assumption The steady state assumption allows us to reduce a large reaction mechanism to a smaller one by eliminating the respective intermediate species.
3.2.1 The reaction H2 + Br2 → 2 HBr The reaction between H2 and Br2 (Bodenstein et al.; Christiansen, Polanyi & Herzfeld) runs as a thermal reaction or as a photochemically induced reaction (initiated by light). In the latter case, it is a chain reaction with a quantum yield of Φ ≈ 106 I
Definition 3.1: Quantum yield Φ: Φ=
I
(3.13)
number of product molecules number of absorbed photons
(3.14)
Reaction mechanism (elementary reactions) describing the H2 -Br2 system: Br2 Br2 + Br + H2 H + Br2 H + HBr Br + HBr
→ → → → → 6→
Br + Br
+M
Br + Br Br + Br HBr + H HBr + Br H2 + Br Br2 + H
→ Br2
chain initiation by thermal dissociation chain initiation by photolysis () chain propagation chain propagation inhibition endothermic (∆ ª = +170 kJ mol) y reaction (6) can be neglected chain termination
(1) (10 ) (2) (3) (4) (6) (5)
(3.15)
3.2 Application of the steady-state assumption
I
44
Rates equations: With the steady state approximations (i.e., [X] ≈ 0) for [Br] and [H], we obtain23 [H] = +2 [Br] [H2 ] − 3 [H] [Br2 ] − 4 [H] [HBr] ≈ 0 y 2 [Br] [H2 ] − 4 [H] [HBr] = +3 [H] [Br2 ] 2 [Br] [H2 ] y [H] = 3 [Br2 ] + 4 [HBr] [Br] = +21 [Br2 ] −2 [Br] [H2 ] + 3 [H] [Br2 ] + 4 [H] [HBr] | {z } [H] =− ≈0 −25 [Br]2 = +21 [Br2 ] − 25 [Br]2 ≈ 0 µ ¶12 1 y [Br] = [Br2 ] 5 µ ¶12 1 2 [H2 ] [Br2 ] × y [H] = 5 3 [Br2 ] + 4 [HBr] [HBr] = 2 [Br] [H2 ] + 3 [H] [Br2 ] − 4 [H] [HBr]
(3.16) (3.17) (3.18) (3.19)
(3.20) (3.21) (3.22) (3.23) (3.24)
Using 2 [Br] [H2 ] − 4 [H] [HBr] = +3 [H] [Br2 ] (Eq. 3.17) and substituting [H] (Eq. 3.18), the last line simplifies to [HBr] = 2 3 [H] [Br2 ] µ ¶12 2 [H2 ] 1 [Br2 ] × = 2 3 [Br2 ] × 5 3 [Br2 ] + 4 [HBr] 2 2 (1 5 )12 [H2 ] [Br2 ]12 = 1 + (4 3 ) [HBr] [Br2 ]
=
I
0 [H2 ] [Br2 ]12 1 + 00 [HBr] [Br2 ]
(3.25) (3.26) (3.27) (3.28)
Result: [HBr] 0 [H2 ] [Br2 ]12 = 1 + 00 [HBr] [Br2 ]
(3.29)
0 = 2 2 (1 5 )12 and 00 = 4 3
(3.30)
with
I
Conclusion: We see that HBr acts as an inhibitor of the total reaction. 23
For the photolysis induced reaction, 1 has to be replaced by 10 , where is the light intensity and 10 the corresponding photolysis rate constants (related to the absorption coefficient).
3.2 Application of the steady-state assumption
45
3.2.2 Further examples I
Exercise 3.1: The thermal decomposition of acetaldehyde according to the overall reaction (3.31) CH3 CHO → CH4 + CO is known to proceed via the following elementary steps (Rice-Herzfeld mechanism): CH3 CHO CH3 + CH3 CHO CH3 CO CH3 + CH3
→ → → →
CH3 + HCO CH4 + CH3 CO CH3 + CO C2 H6
(1) (2) (3) (4)
(3.32)
Derive an expression for the production rate of [CH4 ]. (Hint: Use the steady-state approximations for [CH3 CO] and [CH3 ] and neglect the HCO for the derivation of the expression.) ¤ A more complete mechanism takes into account the thermal decomposition of the HCO radicals according to HCO → H + CO (5) followed by the reaction H + CH3 CHO → H2 + CH3 CO (6). I
Solution 3.1: With the steady state approximations for [CH3 CO] and [CH3 ], we obtain [CH3 CO] = 2 [CH3 ] [CH3 CHO] − 3 [CH3 CO] ≈ 0 2 [CH3 ] [CH3 CHO] y [CH3 CO] = 3
and y
I
[CH3 ] = 1 [CH3 CHO] − 2 [CH3 ] [CH3 CHO] +3 [CH3 CO] − 24 [CH3 ]2 = 1 [CH3 CHO] − 2 [CH3 ] [CH3 CHO] 3 2 [CH3 ] [CH3 CHO] + − 24 [CH3 ]2 3 = 1 [CH3 CHO] − 24 [CH3 ]2 ≈ 0 ¶12 µ 1 [CH3 CHO] y [CH3 ] = 24 [CH4 ] = 2 [CH3 ] [CH3 CHO] µ ¶ 1 [CH4 ] [CH3 CHO]32 = 2 24
(3.33) (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40)
(3.41) ¥
Exercise 3.2: The thermal decomposition of ethane gives mainly H2 + C2 H4 . In addition, small amounts of CH4 are formed. The decay of the C2 H6 concentration can be described by the reaction steps C2 H6 CH3 + C2 H6 C2 H5 H + C2 H6 H + C2 H5
→ → → → →
CH3 + CH3 CH4 + C2 H5 C2 H4 + H C2 H5 + H2 C2 H6
(1) (2) (3) (4) (5)
(3.42)
3.2 Application of the steady-state assumption
I
46
Using the steady-state approximations for the radicals [CH3 ], [C2 H5 ], and [H], derive a rate law describing the overall decay of [C2 H6 ]. Note: From the C-H bond dissociation energies in C2 H6 and C2 H5 , it is known that 3 À 1 . ¤ Solution 3.2: " ¶12 # µ 2 [C2 H6 ] 1 1 3 4 3 [C2 H6 ] − = 1 + + 2 4 5
(3.43) ¥
3.3 Microscopic reversibility and detailed balancing
47
3.3 Microscopic reversibility and detailed balancing The principle of microscopic reversibility is a consequence of the time reversal symmetry of classical and quantum mechanics: If all atomic momenta of a molecule that has reached some product state from an initial state are reversed, the molecule will return to its initial state via the same path. A macroscopic manifestation of the principle of microscopic reversibility is detailed balancing which states that (1) in equilibrium, the forward and reverse rates of a process are equal, i.e., for a 1
reaction A1 À A2 , −1
1 [A1 ] = −1 [A2 ]
(3.44)
and (2) if molecules A1 and A2 are in equilibrium and A2 and A3 are in equilibrium, both with respect to each other, there is also equilibrium between A1 and A3 : 1
2
−1
−2
A1 À A2 À A3 y
(3.45)
3
A1 À A3
(3.46)
−3
Thus, we can write [A2 ] 1 = = 12 [A1 ] −1 [A3 ] 2 = = 23 [A2 ] −2 [A3 ] 3 = = 13 [A1 ] −3 = 12 23 =
(3.47) (3.48) (3.49) 1 2 −1 −2
(3.50)
Detailed balancing allows us to simplify complex reaction systems by eliminating − 1 out of a sequence of reversible reactions that are in equilibrium.
3.4 Generalized first-order kinetics*
48
3.4 Generalized first-order kinetics* 3.4.1 Matrix method* For coupled complex reaction systems with only first-order reactions, the solution of the rate equations can be reduced to an eigenvalue problem (Jost 1974). The solution of the rate equations, i.e., the calculation of the time-dependent concentration profiles, requires finding the eigenvalues and eigenvectors of the rate constant matrix of the system.24 Eigenvalue problems are readily solved in practice using numerical methods (Press 1992). We consider an example which we solved using conventional methods in Section 2.3. I
Example: 1
A1 À A2
(3.51)
2
The rate equations [A1 ] = −1 [A1 ] + 2 [A2 ] [A2 ] = +1 [A1 ] − 2 [A2 ] can be written in a more compact form as a matrix equation: ⎞ ⎛ [A1 ] µ ¶ ¶µ −1 +2 [A1 ] ⎜ ⎟ ⎝ [A ] ⎠ = + − [A2 ] 2 1 2 With the concentration vector A= and the rate constant matrix K= this becomes
µ
µ
[A1 ] [A2 ]
¶
−1 +2 +1 −2
(3.53)
(3.54)
(3.55)
¶
·
A=K·A I
(3.52)
(3.56)
(3.57)
Generalized matrix rate equation for coupled first-order reaction systems: ·
A=K·A 24
A brief overview on matrices and determinants is given in Appendix D.
(3.58)
3.4 Generalized first-order kinetics*
I
49
Ansatz: The ansatz for solving Eq. 3.58 assumes that we can express A in terms of another vector B,25 A=P·B (3.59) where P · B means the dot product, using a rotation matrix P which is defined so that it diagonalizes K via the eigenvalue equation K · P = P · Λ , which by multiplication from the left with P−1 yields P−1 · K · P = Λ (3.60) The eigenvalue matrix Λ is a diagonal matrix of the form ⎛ ⎞ 1 0 0 Λ = ⎝ 0 2 0 ⎠ . 0 0 ..
(3.61)
As defined by Eq. 3.60, its elements on the diagonal {1 2 } are the eigenvalues of the rate constant matrix. The matrix P is called the eigenvector matrix associated with K. The columns of P are the respective eigenvectors associated with the respective eigenvalues . I
Solution: Inserting the ansatz into our matrix equation26 we have or
A=P·B ·
(3.62)
A=K·A
(3.63)
(P · B) =K·P·B
(3.64)
·
P·B=K·P·B
(3.65)
Muliplication of this equation from the left by P−1 gives ·
P−1 · P · B = P−1 · K · P · B or
·
B=Λ·B
(3.66)
(3.67)
This DE can be immediately integrated since Λ is diagonal. The solution is B = Λ B0
(3.68)
© ª where Λ is a diagonal matrix with elements 1 2 , and Λ B0 means element-wise multiplication. The result gives the time dependence of B, i.e., B() starting from the initial values B0 .
25
26
The components of B will be seen to be linear combinations of the components of the concentration vector A. B is immediately obtained once we have the matrix of eigenvectors P (see below). The dot above a concentration vector indicates differentiation by .
3.4 Generalized first-order kinetics*
50
Having B(), we can transform back into the original concentration basis by using B = P−1 · A
(3.69)
B0 = P−1 · A0
(3.70)
P−1 · A = Λ P−1 · A0
(3.71)
and Inserting into our solution for B (Eq. 3.68), we obtain
The final step is now a multiplication from the left with P which yields the solution for Eq. 3.58 A = P · Λ P−1 · A0 (3.72) I
Note: This may look complicated to someone who is not used to it. However, once we have set up K, which is straightforward, everything is done by the computer: The computer computes the • eigenvalues of the rate constant matrix K, • the eigenvector matrix P of the rate constant matrix K, • all necessary inverse matrices, etc., • and does all the matrix multiplications. All you usually have to do is to program the K matrix.
I
Application to our example (solution “by hand”): • Evaluation of the eigenvalue matrix Λ by solution of the eigenvalue equation (3.73)
det (K − Λ · I) = 0 with I as unity matrix:
y
¯ ¯ ¯ −1 − +2 ¯ ¯ ¯ ¯ +1 −2 − ¯ = 0
(3.74)
(−1 − ) · (−2 − ) − (1 · 2 ) = 0 1 2 + 1 + 2 + 2 − 1 2 = 0 ( + (1 + 2 )) = 0
(3.75) (3.76) (3.77)
1 = 0 2 = − (1 + 2 )
(3.78) (3.79)
y
y Λ=
µ
0 0 0 − (1 + 2 )
¶
(3.80)
3.4 Generalized first-order kinetics*
51
• Conclusions: (1) The largest (negative) eigenvalue (2 in this case) determines the shortest time scale and thus the short-time evolution of the reactant concentrations! (2) The smallest (negative) eigenvalue (1 in this case) determines the longest time scale of the reaction, i.e., the evolution of the reactant concentrations at long times. The smallest (negative) eigenvalue thus determines the net reaction rate! (3) Note that in the example the concentrations at long times are constant (equilibrium!) and therefore 1 = 0. • Evaluation of the eigenvectors p1 und p2 by inserting the eigenvalues {1 2 } one by one into K·P=Λ·P (3.81) yields, for each and the respective p , a linear equation of the type K · p = · p
(3.82)
(K − ) · p = 0
(3.83)
(−1 − ) 1 + 2 2 = 0 1 1 + (−2 − ) 2 = 0
(3.84) (3.85)
− 1 1 + 2 2 = 0 1 1 + −2 2 = 0
(3.86) (3.87)
1 1 = 2 2
(3.88)
1 = 2 · 2 = 1 ·
(3.89) (3.90)
y y
• Inserting 1 = 0:
y y
y p1 =
µ
2 · 1 ·
¶
(3.91)
• Inserting 2 = − (1 + 2 ): (−1 + 1 + 2 ) 1 + 2 2 = 0 1 1 + (−2 + 1 + 2 ) 2 = 0
(3.92) (3.93)
2 1 + 2 2 = 0 1 1 + 1 2 = 0
(3.94) (3.95)
y
3.4 Generalized first-order kinetics*
52
y 1 = −2
(3.96)
1 = 2 = −
(3.97) (3.98)
y
y p2 =
µ
−
¶
(3.99)
• Eigenvector matrix (= rotation matrix P): µ ¶ 2 P= 1 −
(3.100)
• The arbitrary factor is determined by the initial conditions to be = 1, so that ¶ µ 2 1 (3.101) P= 1 −1 • Solution for A ():
I
A = P · Λ P−1 · A0
(3.102)
Solution using symbolic algebra programs: • Rate constant matrix K:
K=
µ
−1 +2 +1 −2
¶
• eigenvalues(K) = {−1 − 2 0} : ¶ µ − (1 + 2 ) 0 Λ= 0 0 • eigenvectors(K) =
½µ
−1 1
¶¾
• inverse(P) : P−1
(Ã 1 !) 2 ↔ −1 − 2 ↔0: 1 1 ⎞ ⎛ 2 −1 P=⎝ 1 ⎠ +1 1 ⎛
⎞ 1 2 − ⎜ ⎟ = ⎝ 1+ 2 1 + 2 ⎠ 1 1 1 + 2 1 + 2
(3.103)
(3.104)
(3.105)
(3.106)
3.4 Generalized first-order kinetics*
53
• B0 = P−1 A0 : B0 = P−1 A0 = P−1 ·
µ
10 0
¶
⎞ 1 −10 ⎜ 1 + 2 ⎟ =⎝ ⎠ 1 10 1 + 2 ⎛
(3.107)
• Solution for B (): B = Λ B0
⎛
(3.108)
⎞
1 1 + 2 ⎟ = ⎠ 1 10 1 + 2 ⎞ ⎛ 1 −(1 +2 ) × −10 ⎟ ⎜ 1 + 2 = ⎝ ⎠ 1 10 ×1 1 + 2 µ
−(1 +2 ) 0 0 1
¶
⎜ ⎝
−10
• Solution for A (): Λ
−1
Λ
−1
µ
10 · 0 ¶
(3.109)
(3.110)
¶
A = P · P · A0 = P · P µ µ −( + ) ¶ 10 1 2 0 −1 P · = P· 0 0 1 ⎛ ⎞ µ −( + ) ¶ −10 1 1 2 0 ⎜ 1 + 2 ⎟ = P· ⎝ ⎠ 1 0 1 10 1 + 2 ⎛ µ ¶⎞ 1 −(1 +2 ) −10 ⎜ 1 ¶ + 2 ⎟ ⎜ ⎟ µ = P·⎝ ⎠ 1 1 10 1 + 2 ⎛ ⎞ 2 −(1 +2 ) ⎜ 10 + + 10 1 + ⎟ 1 2 1 2 ⎟ = ⎜ ⎝ 1 −(1 +2 ) ⎠ 10 − 10 1 1 + 2 + 2 ⎞1 ⎛ −(1 +2 ) 2 + 1 ⎟ ⎜ 10 1 + 2 ⎟ = ⎜ −( + ) ⎝ 1− 1 2 ⎠ 10 1 1 + 2
(3.111) (3.112)
(3.113)
(3.114)
(3.115)
(3.116)
This result is identical to the result for [A ()] obtained in Section 2.3.
3.4.2 Laplace transforms* The concept of Laplace and inverse Laplace transforms is extremely useful in chemical kinetics because
3.4 Generalized first-order kinetics*
54
(1) Laplace transforms provide a convenient method for solving differential equations, (2) Laplace transforms form the connection between microscopic molecular properties and statistically (Boltzmann) averaged quantities. I
Definition 3.2: The Laplace transform L [ ()] of a function () is defined as the integral Z∞ (3.117) () = L [ ()] = () − 0
where • is a real variable, • () is a real function of the variable with the property () = 0 for 0, • is a complex variable, • () = L [ ()] is a function of the variable .
I
Definition 3.3: The inverse Laplace transform L−1 [ ()] of the function () is defined as the integral −1
() = L
1 [ ()] = 2
+∞ Z
()
(3.118)
−∞
where • is an arbitrary real constant.
I
Relation between f (t) and F (p): Since L−1 [ ()] recovers the original function (), the pair of functions () and () is said to form a Laplace pair: L−1 [ ()] = L−1 [L [ ()]] = () and
¤ £ L [ ()] = L L−1 [ ()] = ()
(3.119) (3.120)
A list of Laplace and inverse Laplace transforms of some functions can be found in Appendix E (Table E.1).
3.4 Generalized first-order kinetics*
I
55
Solution of differential equations using Laplace transforms: Consider the Laplace transform of the derivative () of a function ():27 L [ () ] =
Z∞
() −
(3.121)
0
= ()
¯∞ − ¯0
Z∞ ¡ ¢ − () −
(3.122)
0
Z∞ = − ( = 0) + () −
(3.123)
= −0 + ()
(3.124)
0
We can solve this equation for () via () =
L [ () ] + 0
and recover the solution () by inverse Laplace transformation, ∙ ¸ −1 L [ () ] + 0 () = L I
(3.125)
(3.126)
Example: Application to two consecutive first-order reactions. As an example, we consider the DE’s for two consecutive first-order reactions
A →1 B →2 C
(3.127)
We already solved this problem previously (in Section 2.6.1) using conventional methods, and repeat the solution now using the Laplace transform method. Towards these ends, in this example, we shall denote the concentrations [A ()] and [B ()] as () and () and their Laplace transforms as () and (). The respective initial concentrations shall be [A]0 = 0 and [B]0 = 0 = 0. Thus we have: • DE for the concentration ():
= 1 0 −1 − 2
(3.128)
• Laplace transform: () can be found using Eq. 3.124: L [] = −0 + ()
Taking the Laplace transform of the r.h.s. of Eq. 3.128:28 £ ¤ L [] = L 1 0 −1 − L [2 ()] £ ¤ 1 0 = 1 0 L −1 − 2 L [ ()] = − 2 () + 1 (Eq. 3.129) = −0 + () 27
28
(3.129)
(3.130) (3.131) (3.132)
We used in line 2: integration by parts; in line 3: the differivative (− ) = − − ; in line 4: the definition () = L [ ()] and the abbreviation ( = 0) = 0 . 1 We used L [ ] = from Table E.1 and L [ ()] = (). −
3.4 Generalized first-order kinetics*
56
Since we have 0 = 0 from the initial condition, this becomes 1 0 + 1
(3.133)
1 1 + 1 + 2
(3.134)
() + 2 () = • Solution for ():
() = 1 0
• Inverse Laplace transform:29 () = L [ ()] =
¢ 1 0 ¡ −1 − −2 2 − 1
which is the same as obtained in Section 2.6.1.
29
−1
From Table E.1, we see that L
∙
¸ ¢ ¡ 1 1 = − . ( − ) ( − ) ( − )
(3.135)
3.5 Numerical integration
57
3.5 Numerical integration I
Initial value problems: For complex reaction systems, we have to integrate the coupled nonlinear DE systems numerically using the computer. We start with the known · initial values of the concentrations at time 0 and the associated DE’s ( ) and want to compute the values of at time 0 . In numerics, these problems are known as initial value problems. With additional conditions, for example for the spatial distributions of the concentration (fixed values at (0 0 0 1 )), they become boundary value problems. In solving these problems, we generally have to deal with systems of stiff nonlinear coupled (ordinary) differential equations (DE systems are said to be stiff, if the rate constants (more precisely: the eigenvalues) vary by many orders of magnitude; under these conditons, the changes of the dependable variables differ by many orders of magnitude). Thus we have to solve • equations (kinetic equations + transport processes), with • variables (concentrations), and the given • initial values and boundary values.
I
Applications of numerical integration methods: • Combustion chemistry: — Stationary combustion (for example diffusion flames), — Instationary combustion (ignition processes, turbulent combustion) - first successful attempts have been made, — Problem: For all but the simplest fuels, we have to deal with hundreds of species and thousands of elementary reactions. • Atmospheric chemistry: — 2D “box” models ( ( )) are straightforward. — 4D models are required (time, latitude, longitude, height)! — Huge problems, when it comes to ∗ feedback with oceans, etc.? ∗ do we know all reactions? ∗ heterogeneous reactions? • Physical organic chemistry: — Elucidation of reaction mechanisms. • Macromolecular chemistry (including industrial macromolecular synthesis): — Prediction of chain length and other properties of polymers and co-polymers.
3.5 Numerical integration
I
58
Survey of methods of numerical integration methods: • explicit methods (e.g., Runge-Kutta), • implicit methods (e.g., Gear), • extrapolation methods (e.g., Richardson, Bulirsch-Stoer, Deuflhard). In the following, we consider the basics of the most convenient methods.
3.5.1 Taylor series expansions The Taylor expansion is the basis for all numerical integration methods. We start from the initial value of (0 ) at some time 0 (0 ) = 0
(3.136)
and want to determine the value (0 + ) at time 0 + . Using the DE for ·
() = ( )
(3.137)
we expand in a Taylor series around 0 : 2 ·· · (0 0 ) + 2! 2 · · (0 0 ) + = (0 ) + · (0 0 ) + 2! ·
(0 + ) = (0 ) + · (0 0 ) +
(3.138) (3.139)
Recursion formula: +1 = + · ( ) +
2 · · ( ) + 2!
(3.140)
The first-order term ( ) is given by the rate equation at , the second-order ·
term ( ) (which is taken into account for example by the 2 order Runge-Kutta method) and higher order terms (⇒ higher order Runge-Kutta methods) have to be evaluated numerically by finite differences.
3.5.2 Euler method The Euler method is based on a 1 order Taylor expansion. If the step size is small enough, the higher order terms can be neglected: ·
(3.141)
= ( )
(3.142)
(0 + ) = (0 ) + · (0 ) + O(2 ) Initial value problem:
·
3.5 Numerical integration
59
with (0 ) = 0
(3.143)
·
Result by substituting (0 ) = (0 ): ¡ ¢ (0 + ) = (0 ) + (0 ) + O 2
(3.144)
Recursion formula (see Fig. 3.1):
+1 = + +1 = + ( )
(3.145) (3.146)
Problem: Large errors for practical step sizes.
I
Figure 3.1: Euler method.
3.5.3 Runge-Kutta methods I
2 order Runge-Kutta method: Initial value problem:
= ( )
·
(3.147)
(0 ) = 0
(3.148)
with
3.5 Numerical integration
60
2 order Runge-Kutta ansatz = Taylor expansion to 2 order: 2 ·· (0 ) + O(3 ) (3.149) 2! Ã ! · · 2 (0 + ) − (0 ) · + O(3 ) (3.150) ≈ (0 ) + (0 ) + 2 ´ ³· · = (0 ) + (0 + ) + (0 ) + O(3 ) (3.151) 2 | {z } ·
(0 + ) = (0 ) + (0 ) +
=2× slope of () at midpoint
Recursion formula:
+1 = + +1 = + 2 + O(3 ) 1 = ( ) µ ¶ 1 2 = + + 2 2
I
Figure 3.2: Illustration of the 2 order Runge-Kutta method.
(3.152) (3.153) (3.154) (3.155)
3.5 Numerical integration
61
I
4 order Runge-Kutta method: In practice, one frequently uses the 4 order Runge-Kutta method.
I
Figure 3.3: Illustration of the 4 order Runge-Kutta method.
3.5.4 Implicit methods (Gear): The Runge-Kutta method becomes unstable for stiff DE systems. In this case, it is preferable to employ implicit (also called iterative or backward) integration methods. The starting point is the implicit form of the Euler method, ·
(0 + ) = (0 ) + · (0 + ) + O(2 )
(3.156)
·
For the initial value problem = ( ) with (0 ) = 0 this leads to the recursion formula: +1 = + +1 = + (+1 +1 )
(3.157) (3.158)
Using the so-called predictor-corrector methods, +1 is predicted in a first step by starting from using one of the explicit methods described above. The obtained trial (0) value +1 is then corrected by using a backward integration scheme (Gear 1971a, Gear (1) (1) 1971a, Press 1992) to obtain a better estimate +1 . Then, and +1 are used for (2) a new prediction-corrrection cycle, giving +1 . The procedure is repeated until the difference between successive iterations falls below some tolerance limit . The predictor-corrector method of Gear has been the method of choice for a long time (Gear 1971a, Gear 1971a).
3.5 Numerical integration
62
3.5.5 Extrapolation methods (Bulirsch-Stoer) The idea behind extrapolation methods is that the final answer for +1 is approximated by an analytical function in the step size (Richardson extrapolation). The best strategy is to use a rational extrapolating function as the ansatz, and not a polynomial function (Bulirsch-Stoer). This function is determined and fitted to the problem of interest by performing calculations with various values of . The extrapolating function is then used for extrapolating from to +1 . The optimal strategy for stiff DE systems is to follow the Bulirsch-Stoer method (Stoer 1980) in the form implemented by Deuflhard (Bader 1983, Deuflhard 1983, Deuflhard 1985). For Fortran subroutines, see (Press 1992).
I
Figure 3.4: The principle of extrapolation methods.
3.5.6 Example: Consecutive first-order reactions As an example, we solve the DE’s for two consecutive first-order reactions
1 2 A −→ B −→ C
(3.159)
(a) exactly, using symbolic algebra software like MathCad, Maple, Mathematica, or MuPad (the math engine built-in into SWP, the program used to write this script) and (b) using numeric integration.
3.5 Numerical integration
I
63
Exact solution using symbolic algebra software: (1) To set-up the initial value problem for the program,30 we enter the following a 1 × 6 matrix: ⎡ ⎤ 0 = −1 ⎢0 = +1 − 2 ⎥ ⎢ ⎥ ⎢ 0 = +2 ⎥ (3.160) ⎢ ⎥ ⎢ (0) = 0 ⎥ ⎣ ⎦ (0) = 0 (0) = 0 (2) To obtain the solution, we select the menu item "SolveODE, Exact". The program gives the output: Exact solution is: ⎡ µ ¶⎤ −2 1 −1 2 ⎢ () = 0 1 − 0 1 2 − + 0 1 − ⎥ 1 2 1 2 1 ⎥ ⎢ −1 −2 ⎥ ⎢ ⎥ ⎢ () = − 0 1 0 1 ⎥ ⎢ − − 2 1 2 1 ⎥ ⎢ µ ¶ − − 1 1 ⎦ ⎣ 1 2 0 1 2 () = − 0 1 1 2 − 1 2 − 1
(3.161)
(3) After simplifying, factorizing, and rewriting the output, we obtain ⎡ ¢⎤ 0 ¡ () = 2 − 1 + 1 −2 − 2 −1 ⎢ ⎥ 2 − 1 ⎢ ⎥ ¢ 1 ¡ −2 −1 ⎢ ⎥ − () = 0 ⎣ ⎦ 1 − 2 −1 () = 0
(3.162)
We immediately recognize these results from section 2.
I
Numerical solutions: Numerical integration is performed by the different symbolic algebra programs using one of the methods outlined in the preceding subsection. We have to specify the DE’s, the initial values, and the values for the rate constants.31 (1) We adopt the rate constant values 1 = 100 and 2 = 100. (2) We define the problem (now using capital letters): ⎡
⎤ 0 = − ⎢ 0 = + − 10 ⎥ ⎢ ⎥ ⎢ 0 = +10 ⎥ ⎢ ⎥ ⎢ (0) = 1 ⎥ ⎣ (0) = 0 ⎦ (0) = 0 30 31
(3.163)
The different software packages differ in the ways in which the problems have to be set up. SWP can handle the problem only if the rate constant values are explicitly entered in the DE’s. Other software does better.
3.5 Numerical integration
64
, Functions defined:
y 1.0
0.5
0.0 0
2
4
(3) Application of "SolveODE, Numeric" gives the output: , Functions defined: This means that the program now “knows” these functions. (4) We can plot , , and using the Plot2D menu point: 1
c 0.75
0.5
0.25
0 0
1.25
2.5
3.75
5
t
(We add the plots for and by dragging both symbols to the plot window) (5) With 0 = 1, 1 = 1, and 2 = 10, we also plot the exact solutions , , and =
¢ 0 ¡ 2 − 1 + 1 −2 − 2 −1 2 − 1 = 0
¢ 1 ¡ −2 − −1 1 − 2 = 0 −1
(3.164) (3.165) (3.166)
3.5 Numerical integration
65
1
c 0.75
0.5
0.25
0 0
1.25
2.5
3.75
5
t
(6) The rest is just some fixing up: We combine the two plots, with an increased number of points plotted, add axis labels and colors: 1
c(t) 0.75
0.5
0.25
0 0
1.25
2.5
3.75
5
t
Numerical solution of the DE system defined by Eq. 3.163. Full lines: numeric solutions. Dotted lines: exact solutions.
The exact and the numerical solutions are identical within numeric precision.
3.6 Oscillating reactions*
66
3.6 Oscillating reactions* The reactant and product concentrations normally approach equilibrium in a smooth (often exponenential) fashion. Under special circumstances, however, the concentrations may also show oscillations and/or bistability as they approach equilibrium. Such behavior can occur only if there is a chemical feedback by autocatalysis.
3.6.1 Autocatalysis Example for autocatalysis:
A+B→B+B Rate law: −
[A] = [A] [B]
(3.167)
(3.168)
To solve this DE, we substitute for [B ()] using the mass balance relation [A]0 −[A ()] = [B ()] − [B]0 and introduce = [A ()] as reaction progress variable: [B ()] = [A]0 + [B]0 − [A ()] = [A]0 + [B]0 −
(3.169) (3.170)
This gives the DE = ([A]0 + [B]0 − ) Z Z = − ([A]0 + [B]0 ) − 2 −
y
Solution:
or
¶ µ [A]0 [B ()] 1 = − ln [A]0 + [B]0 [B]0 [A ()] [B ()] =
[A]0 + [B]0 1 + ([A]0 [B]0 ) exp [− ([A]0 + [B]0 ) ]
(3.171) (3.172)
(3.173)
(3.174)
3.6 Oscillating reactions*
I
67
Figure 3.5: Autocatalysis.
• As can be seen from the sigmoid (-shaped) concentration-time profile of [B] in Fig. 3.5, the rate of increase of [B] increases up to the inflection point ∗ . • This curve is typical for a population increase from a low plateau to a new (higher) plateau after a favorable change of the environment due to, e.g., the availability of more food, a technical revolution, a reduction of mortality by a medical breakthrough, . . . Many interesting games which can be played based on these ideas are described in (Eigen 1975).
3.6.2 Chemical oscillations The presence of one or more autocatalytic steps in a reaction mechanism can lead to chemical oscillations. a) The Lotka mechanism The origin of chemical oscillations can be understood by considering the reaction mechanism proposed by Lotka in 1910 (see Lotka 1920):
A + X →1 X + X X + Y →2 Y + Y Y →3 Z
(1) (2) (3)
(3.175)
3.6 Oscillating reactions*
68
• Both reactions (1) and (2) are autocatalytic. • The Lotka mechanism illustrates the principle, but it does not correspond to an existing chemical reaction system. • A real oscillating reaction is the Belousov-Zhabotinsky reaction; this reaction may be described using a more complicated mechanism (see below). In order to model the resulting chemical oscillations, we assume that the reaction takes place in a flow reactor, which is constantly supplied with new A such that the concentration of A stays constant ([A] = [A]0 ), while the product Z is constantly removed. I
Rate equations and steady-state solutions: [X] = +1 [A]0 [X] − 2 [X] [Y] [Y] = +2 [X] [Y] − 3 [Y]
(3.176) (3.177)
Steady-state solutions: [X] = +1 [A]0 [X] − 2 [X] [Y] = 0 [Y] = +2 [X] [Y] − 3 [Y] = 0
(3.178) (3.179)
Dividing these equations by [X] and [Y], respectively, we find 2 [Y] = 1 [A]0 2 [X] = 3
(3.180) (3.181)
Since [A] = [A]0 , the steady state solutions for [X] and [Y] are independent of time! I
Displacements from steady-state solutions: What happens if the concentrations are displaced from the steady-state values by small amounts and ? (1) Ansatz: [X] = [X] + [Y] = [Y] +
(3.182) (3.183)
3.6 Oscillating reactions*
69
(2) Rate equations: = +1 [A]0 ([X] + ) − 2 ([X] + ) ([Y] + ) = +1 [A]0 [X] + 1 [A]0 − 2 [X] [Y] | {z }
(3.184)
=1 [A]0 [X]
−2 [X] − 2 [Y] − 2 | {z } =1 [A]0
= −2 [X] − 2
= +2 ([X] + ) ([Y] + ) − 3 ([Y] + ) = 2 [X] [Y] + 2 [X] + 2 [Y] +2 − 3 [Y] − 3 |{z} |{z} =2 [X]
(3.185) (3.186)
=2 [X]
= +2 [Y] + 2
(3.187)
(3) Simplified DE’s for small displacements: We may neglect the terms containing . Thus, we obtain two coupled first-order DE’s = −2 [X] = +2 [Y]
(3.188) (3.189)
which can be converted to a single second-order DE by the ansatz ·
(3.190)
·
(3.191)
= − = +
Second-order DE:
··
·
= − = −
(3.192)
Formal solution of this second-order DE: () = [X ()] − [X] = 1 + 2 −
(3.193)
(4) Eigenvalues: To determine the eigenvalue belonging to the linear DE’s, we have to solve the determinant ¯ ¯ ¯ ¯ ¯0 − − ¯ ¯ −2 [X] ¯¯ ¯=¯ ¯ (3.194) ¯ + 0 − ¯ ¯2 [Y] ¯=0 y
2 + 2 [X] [Y] = 0
(3.195)
y The solutions for are purely imaginary: ¡ ¢12 12 = ± 22 [X] [Y] = ± (1 3 [A]0 )12
(3.196) (3.197)
3.6 Oscillating reactions*
70
(5) Solution for (): () = 1 + 2 − = 0 cos
(3.198)
= (1 3 [A]0 )12
(3.199)
with
This means that if the system is displaced from its steady-state, the species concentrations will start to oscillate and the displacements may even grow in magnitude until the amplitude reaches the limiting value 0 (see limit cycle diagram in Fig. 3.6).
I
Figure 3.6: Limit cycle diagram for chemical oscillations according to the Lotka mechanism.
b) The Brusselator32 Reaction scheme: A B+X Y+2 X X 32
→1 →2 →3 →4
X R+Y 3X S
(1) (2) (3) (4)
(3.200)
The name ‘Brusselator’ stems from the workplace of its inventor Prigogine (Brussels). The model does not correspond to a real chemical system.
3.6 Oscillating reactions*
71
Here, A and B are reactants, R and S are final products, and X and Y are intermediates. The autocatalytic reaction is reaction (3). Rate equations (using “dimensionless time” = 4 ) and steady-state concentrations: · [X] 1 [A] − 2 [B] [X] + 3 [X]2 [Y] − 4 [X] = [X] = 4 2 · [X] 2 [B] [X] − 3 [X] [Y] = [Y] = 4
(3.201) (3.202)
Steady-state concentrations: 1 [A] 4 2 4 [B] = 1 3 [A]
[X] =
(3.203)
[Y]
(3.204)
We now investigate the effect of small displacements from steady-state by [X] and [Y]. The reaction rates respond to these displacements according to ⎛ · ⎞ ⎛ · ⎞ ⎝ [X] ⎠ [X] and ⎝ [Y] ⎠ [Y] (3.205) [X] [Y]
y
• The steady-state is stable if ⎛
·
⎞
⎛
·
⎞
⎝ [X] ⎠ + ⎝ [Y] ⎠ 0 [X] [Y]
(3.206)
• The steady-state is unstable if ⎛ · ⎞ ⎛ · ⎞ ⎝ [X] ⎠ + ⎝ [Y] ⎠ ≥ 0 [X] [Y]
(3.207)
• Stability criterion: ⎛ · ⎞ ⎛ · ⎞ 2 2 ⎝ [X] ⎠ + ⎝ [Y] ⎠ = 2 [] − 1 3 [] − 1 [X] [Y] 4 42
(3.208)
c) Belousov-Zhabotinsky reaction Oxidation of malonic acid to CO2 by bromate, catalyzed by cerium ions in acidic solution: + 2BrO− 3 + 3CH2 (COOH)2 + 2H −→ 2BrCH(COOH)2 + 3CO2 + 4H2 O
(3.209)
The reaction is probed by addition of the redox indicator ferroin/ferriin: The color varies between red and blue depending on the concentrations of Ce4+ and Ce3+
3.6 Oscillating reactions*
I
Figure 3.7: Mechanism of the Belousov-Zhabotinsky reaction.
I
Figure 3.8: Phases I - III of the Belousov-Zhabotinsky reaction.
72
3.6 Oscillating reactions*
I
73
Figure 3.9: Key steps of the Belousov-Zhabotinsky reaction.
d) The Oregonator33 The Oregonator model can be used to describe the Belousov-Zhabotinsky reaction. I
Schematic model and kinetic equations:
→1 →2 →3 →4
X+R (1) 2R (2) 2 X + 2 Z (3) A+R (4) 5 1 B+Z→ Y (5) 2
A+Y X+Y A+X 2X
(3.210)
with A = BrO− 3 B = CH2 (COOH)2 + BrCH(COOH)2 R = HOBr X = HBrO2 − Y = Br Z = Ce4+ . Using dimensionless time ( = 5 [B] ), we obtain
33
The name ‘Oregonator’ stems from the workplace of its inventor Noyes (University of Oregon).
3.6 Oscillating reactions*
[X] 1 [A] [Y] − 2 [X] [Y] + 3 [A] [X] − 24 [X2 ] = 5 [B] 1 − 5 [B] [Z] [A] [Y] − [X] [Y] + 1 2 [Y] 2 = 5 [B] [Z] 23 [A] [X] − 5 [B] [Z] = 5 [B]
I
74
(3.211)
(3.212) (3.213)
Figure 3.10: Numerical integration of the Oregonator model (rate constants in l mol−1 s−1 : 1 = 0005, 2 = 10, 3 = 10, 4 = 10, 5 = 0004; initial concentrations in mol l−1 : [A] = 001, [B] = 10, [X]0 = 00, [Y]0 = 00, [Z]0 = 000025.)
3.6 Oscillating reactions*
75
3.7 References Bader 1983 G. Bader, P. Deuflhard, Numerische Mathematik 41, 373 (1985) Deuflhard 1983 P. Deuflhard, Numerische Mathematik 41, 399 (1985). Deuflhard 1985 P. Deuflhard, SIAM Review 27, 505 (1985). Eigen 1975 M. Eigen, R. Winkler, Das Spiel, Piper, München, 1975. Gear 1971a C. W. Gear, Commun. Am. Comput. Mach. 14, 1976 (1971). Gear 1971b C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, 1971. Houston 1996 P. L. Houston, Chemical Kinetics and Reaction Dynamics, McGrawHill, Boston, 2001. Jost 1974 W. Jost, in Physical Chemistry: An Advanced Treatise, Ed.: W. Jost, Vol. VIa, Academic Press, New York, 1974. Lotka 1920 A. J. Lotka, Undamped Oscillations Derived from the Law of Mass Action, J. Am. Chem. Soc. 44, 1595 (1920). Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1992. Scott 1994 S. K. Scott, Oscillations, Waves, and Chaos in Chemical Kinetics, Oxford Science Publications, Oxford, 1994. Stoer 1980 J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer, New York, 1980.
3.7 References
76
4. Experimental methods In this section we consider experimental techniques for determining rate coefficients. We’ll get to faster reactions as we go down the list. Special techniques used in quantum state-resolved reaction dynamics, photodissociation, or transition state spectroscopy (molecular beams, state specific detection, photofragment translational spectroscopy, photofragment imaging, other pump-probe techniques, fluorescence up-conversion, fs-CARS, fs-DFWM) will be considered later (in advanced courses), but we’ll briefly touch photochemistry and “femtochemistry”.
4.1 End product analysis • first step for setting up reaction mechanisms, • all convenient analytical techniques (HPLC, GC, GC-MS, MS, UV, FTIR, . . . ), • Problem: No “direct” kinetic information because of lack of time resolution.
I
Figure 4.1: FTIR spectrometer for kinetic studies of photolysis-induced reactions.
4.2 Modulation techniques
77
4.2 Modulation techniques • especially for photo-induced reactions. • photolysis light has to be modulated at a frequency that is comparable to that of the reaction y time scale of 10’s of ms and longer. • kinetic information is derived from the “in-phase” and the “in-quadrature” components of the reactant and product concentrations as function of modulation frequency.
I
Figure 4.2:
4.3 The discharge flow (DF) technique
78
4.3 The discharge flow (DF) technique DF technique is used for gas phase reactions on ms timescale: • laminar flow along tubular reactor: ∆ =
∆
(4.1)
with = mean flow speed, ∆ = distance along the flow tube. • y first-order reaction gives exponential decay of concentration along ∆. • in reality ⇒ Hagen-Poisseuille flow (parabolic flow profile), • but fast radial diffusion of reactants (at pressures of 1 to 10 mbar) gives “plug shaped” radial concentration profiles despite parabolic flow profile. • more complicated, when “back diffusion” and radial diffusion have to be taken into account (at higher pressures). • problem: heterogeneous reactions lead to first-order decay of reactive radicals even in the absence of a reactant. Combinations with numerous highly sensitive detection techniques, for instance: • mass spectrometry (MS): universal, but often low sensitivity; problems with fragmentation of molecules in the ion source, • laser induced fluorescence (LIF): excitation to electronically excited state followed by relaxation by spontaneous emission. very sensitive (e.g., for OH (2 Σ ← 2 Π) detection limit below 106 molecules cm3 ), • electron spin resonance (ESR): for paramagnetic atoms by exploiting the Zeeman effect depending on magnetic field = 0 +
(4.2)
∆ = 1 y ∆ =
(4.3)
ESR transitions are in the microwave region (X band: 9 GHz). • laser magnetic resonance (LMR): very sensitive; based on observation of rotational transitions between Zeeman-shifted rotational levels of paramagnetic radicals (rot = rotational constant). 00 = rot ( + 1) + 00 00 0 = rot ( + 1) + 0 0
(4.4) (4.5)
∆ = ±1 ∆ = 0 ±1 : y 0 = rot ( 00 + 1) ( 00 + 2) + 0 0 y ∆ = = 2rot ( 00 + 1) + (0 0 − 00 00 )
(4.6) (4.7) (4.8)
LMR transitions are in the FIR region (100 m ≤ ≤ 2000 m).
4.3 The discharge flow (DF) technique
I
Figure 4.3: Schematic diagram of a DF/MS combination.
I
Figure 4.4: Schematic diagram of a DF/LIF combination.
79
4.3 The discharge flow (DF) technique
80
I
Figure 4.5: Principles of Electron Spin Resonance (ESR) and Laser Magnetic Resonance (LMR).
I
Figure 4.6: Laser Magnetic Resonance (LMR).
4.3 The discharge flow (DF) technique
81
I
Figure 4.7: Schematic diagram of a DF/LMR combination
I
Figure 4.8: Flow reactor with laser flash photolysis and time-resolved mass spectrometric detection of transient reactants and products.
4.4 Flash photolysis (FP)
82
4.4 Flash photolysis (FP) • “pump-probe” technique: — generation of transient species by flash photolysis using fast flash lamp or pulsed lasers (“pump”), — fast detection of transient species by UV absorption spectrometry, laser induced fluorescence (LIF), cavity ringdown spectroscopy (CRDS) (”probe”). • for gas and liquid phase reactions on nanosecond timescale (Norrish & Porter).
I
Figure 4.9: Typical “Pump-Probe” set-up for kinetics studies by pulsed laser photolysis and detection by laser-induced fluorescence.
I
Cavity Ringdown Spectroscopy (CRDS): • CRDS is an ideal method for absorption spectroscopy with pulsed lasers despite of their large (±5 %) amplitude fluctuations. • The absorption measurement is transformed into the time domain: — time dependence of signal transmitted through a resonantor: = 0 exp (− )
(4.9)
4.4 Flash photolysis (FP)
83
— decay of intensity for pass through cavity (length , mirror reflectivity , absorption path length ): µ ¶ 1− =− − (4.10) {z } {z } | | due to reflection losses at mirrors
due to absorption
— with y
=
(4.11)
= − [(1 − ) + ]
(4.12)
— empty cavity ( = 0): (1 − )
(4.13)
[(1 − ) + ]
(4.14)
−1 0 = — with absorber ( 6= 0): −1 =
— The difference of the ringdown times measures the absorption coefficient and is related to absorber concentration [A] by −1 − −1 0 = = [A]
(4.15)
• Performance of a CRD spectrometer: — mirrors: = 99999%
(4.16)
— measured empty cavity decay time: 0 ≈ 300 s
(4.17)
— effective absorption pathlength (1 m cell): eff = 90 km
(4.18)
• applications to chemical kinetics: — slow reactions: pump-probe scheme with variable time delay. first-order reaction with rate constant 1 : [A] = [A]0 −1 y
¡ ¢ ln −1 − −1 = − 1 0
(4.19) (4.20)
— fast reactions: kinetics and ringdown occur on the same time scale. y (extended) Kinetics and Ringdown model ((e)SKaR); see Brown2000 and Guo2003 for details.
4.4 Flash photolysis (FP)
I
Figure 4.10:
I
Figure 4.11:
84
4.5 Shock tube studies of high temperature kinetics
85
4.5 Shock tube studies of high temperature kinetics • Detection of atoms by atomic resonance absorption spectrometry (ARAS), • Detection of small radicals by frequency modulation absorption spectrometriy (FM-AS) I
Figure 4.12:
I
Figure 4.13:
4.6 Stopped flow studies
86
4.6 Stopped flow studies • the analogue of the flow tube technique for liquid phase reactions. • applications mainly kinetics of enzyme reactions, studies of protein folding dynamics
I
Figure 4.14:
4.7 NMR spectroscopy
4.7 NMR spectroscopy • line coalescence • good for interconversion of two conformers (“isomerization”)
I
Figure 4.15:
87
4.8 Relaxation methods
88
4.8 Relaxation methods • application to fast reactions in the liquid phase (Eigen) • -jump, -jump, -jump, . . . : (1) ⇒ displacement from equilibrium
(2) ⇒ measurements of relaxation into new equilibrium state by transient absorption spectroscopy, conductivity, . . .
4.9 Femtosecond spectroscopy
89
4.9 Femtosecond spectroscopy
I
Figure 4.16: Femtosecond pump-probe spectroscopy for the investigation of ultrafast reactions (time resolution down to the order of 10 fs).
I
Figure 4.17: Signals in broadband femtosecond transient absorption spectroscopy.
4.9 Femtosecond spectroscopy
90
I
Figure 4.18: Broadband femtosecond transient absorption spectroscopy of photochromic molecular switches in SFB 677.
I
Figure 4.19: Broadband femtosecond transient absorption spectroscopy of Disperse Red 1.
4.9 Femtosecond spectroscopy
91
I
Figure 4.20: Broadband femtosecond transient absorption spectroscopy of Furyl Fulgides.
I
Figure 4.21: Ultrafast Electronic Deactivation in DNA Building Blocks.
4.9 Femtosecond spectroscopy
92
I
Figure 4.22: Femtosecond fluorescence up-conversion data for DNA strands.
I
Figure 4.23: Ultrafast Electronic Deactivation in DNA Building Blocks: d(pApA) as Model System.
4.9 Femtosecond spectroscopy
93
4.10 References Busch 1999 K. W. Busch, M. A. Busch, Cavity-Ringdown Spectroscopy, ACS Symposium Series, Vol 720, American Chemical Society, Washington, 1999. Brown 2000 S. S. Brown, A. R. Ravishankara, H. Stark, J. Phys. Chem. A 104, 7044 (2000). Friebolin 1999 H.Friebolin, Ein- und zweidimensionale NMR-Spektroskopie, WileyVCH, Weinheim, 1999. Guo 2003 Y. Guo, M. Fikri, G. Friedrichs, F. Temps, An Extended Simultaneous Kinetics and Ringdown Model: Determination of the Rate Constant for the Reaction SiH2 + O2 . Phys. Chem. Chem. Phys. 5, 4622 (2003). OKeefe 1988 A. O’Keefe, D. A. G. Deacon, Rev. Sci. Instrum. 59, 2544 (1988).
4.10 References
94
5. Collision theory So far, we have considered more or less complex chemical reactions following more or less complicated rate laws, but we have only considered the rate coefficients appearing in the rate laws as empirical quantities, which had to be determined experimentally. We shall now ask the question, whether we can rationalize and predict those rate coefficients using theoretical models. Towards these ends, we have to look at the reactions at a microscopic, molecular level. In particular, we have to • consider the relation between the microscopic (individual molecule) properties and the macroscopic (thermally averaged) rate quantities (rate constants, product branching ratios) • develop models for the microscopic (molecular level) progress of individual elementary reactions, including the different molecular degrees of freedom (quantum states). • develop sufficiently simplified models averaging over the molecular degrees of freedom which still allow us to make practically useful, relevant, and sufficiently accurate predictions. We shall do so in the next four chapters. I
Collision theory: To begin with, this chapter considers bimolecular reactions in the gas phase. These obviously require a collision between two molecules as condition for a reaction to take place. • We start with the hard sphere collision model of structureless particles as the simplest model for bimolecular reactions. Although this model has two major flaws, it leads to the right result because of a compensation of errors. • We then consider results of kinetic gas theory, which allow us to properly take into account the molecular speed distributions. • We apply the hard sphere collision model to understand transport processes in the gas phase (diffusion, heat conductivity, viscosity). • We then proceed with a correct derivation of advanced collision theory models starting from differential cross sections (which can be measured in crossed molecular beam experiments). Advanced collision theory is the method of choice for fast gas phase reactions which can be studied in detail by molecular beam experiments and handled by quantum theory.
5.1 Hard sphere collision theory
95
5.1 Hard sphere collision theory 5.1.1 The gas kinetic collision frequency I
The gas kinetic collision frequency as upper limit for rate constants of bimolecular reactions: The gas kinetic collision frequency constitues an upper limit for the rate constant of bimolecular gas phase reactions. We assume that the molecules A and B are structureless hard spheres (radii , ) moving around with fixed mean speeds and and look at the number of collisions between A and B per unit time.
I
Figure 5.1: Derivation of the hard sphere gas kinetic collision frequency.
I
Discussion of the molecular quantities affecting the rate constant: • collision parameter (= projected distance between centers of gravity):
(5.1)
• maximal collision parameter = max value leading to a collision max : ≤ max = + =
(5.2)
• collision cross section = 2 (= reactive cross section)34 : = 2 = ( + )2 34
Deutsch: reaktiver Wirkungsquerschnitt.
(5.3)
5.1 Hard sphere collision theory
96
• mean speed of A and B:
r
r 8 8 = = r r 8 8 = =
• mean relative speed (2 different molecules, i.e., A 6= B):35 r 8 = with the reduced mass =
+
(5.4) (5.5)
(5.9)
(5.10)
• mean relative speed (single type of molecules, i.e., A = B): = y
I
1 = + 2 √ = 2×
r
8
(5.11)
(5.12)
Hard sphere gas kinetic collision frequency: • We want to determine the number of collisions of a single molecule A with molecules B in time . This is given by the product of the (volume that A has swept in ) × (number density of B): — volume of cylinder that A has swept in time : volume = cross section ×
(5.13)
2 ×
(5.14)
y
35
The mean relative speed of two molecules with velocities u and u is found by looking at the square of the difference speed 2 = |u − u |2 = | |2 + | |2 − 2 | | | | cos
(5.6)
The third term averaged over cos is 0, therefore 2 = | |2 + | |2 =
3 3 3 ( + ) 3 + = =
(5.7)
³ ´12 differ only by a constant factor, we also find that Since and 2 =
s
8
(5.8)
5.1 Hard sphere collision theory
97
— number density of B (5.15)
— Number of collisions of a single A in time : ⇒ number of molecules B in that volume = 2 × × (5.16) • Number of all collisions between A and B: = 2 × ×
(5.17)
• Modification for collisions between identical molecules (collisions must not be counted twice): 1 × 2 × × 2√ 2 = × 2 × × 2
(5.18)
=
(5.19)
• Comparison with rate expression: −
=− = × =
(5.20)
y • Expression for rate constant for collisions A + B: = 2 × = × max
y max
2
= ×
µ
8
(5.21)
¶12
(5.22)
• Modification for collisions between identical molecules (A + A): max
1 1 = 2 × = √ 2 × 2 2
µ
8
¶12
(5.23)
These expressions give the upper limits for the rate constants of bimolecular reactions. The upper limits are reached only if • there are no energetic restrictions (no energy barriers), • all collisions lead to reaction (no steric constraints), • molecules can be approximated as hard spheres (unrealistic, usually one has ), • all molecules collide with the same relative speed (in reality there is a distribution of speeds).
5.1 Hard sphere collision theory
I
98
Mean free pathlength: • The mean free pathlength is the distance that a molecule flies between two successive collisions: =
distance/time distance = = # of collisions # of collisions/time
y =
2
1 1 ∝ ×
(5.24)
(5.25)
• Modification for collisions between identical molecules (A + A): 1 1 = √ 2 2 × I
Time between two collisions: 1 1 = 2 ×
I
(5.26)
(5.27)
Example 5.1: Collision frequency for O2 - N2 @ = 298 K = 1000 mbar: Molecular parameters: 2 = 22 = 3467 Å 2 = 22 = 3798 Å y = 36 Å = 36 × 10−10 m 2
= 2 = 41 Å 1 28 × 32 × 10−3 kg mol × = 28 + 32 s 8 = 650 m s =
(5.28) (5.29) (5.30) (5.31) (5.32) (5.33)
(but (O2 ) = 444 m s)
(5.34)
µ
(5.35)
Rate constant: max
¶12 8 = × −10 = 26 × 10 cm3 s = 16 × 1014 cm3 mol s 2
(5.36) (5.37)
Mean free pathlength: = = = = =
1000 mbar 40 × 10−5 mol cm3 25 × 1025 m3 36 × 10−10 m 1 × 10−7 m
(5.38) (5.39) (5.40) (5.41) (5.42)
5.1 Hard sphere collision theory
99
Rule-of-thumb: = 03 mbar y = 03 mm
(5.43)
= 1000 mbar 1 1 = 2 × = 15 × 10−10 s ≈ 01 ns
(5.44)
Time between collisions:
(5.45) (5.46) (5.47) ¤
5.1.2 Allowance for a threshold energy for reaction I
The threshold energy E0 : The majority of reactions exhibit a threshold energy, below which the reaction does not occur (quantum mechanical tunneling is excluded here). The molecules must have a minimum energy min to overcome this reaction threshold energy 0 . For our structureless spherical molecules, we consider only translational energy :36 0 : no reaction 0 : reaction
(5.48) (5.49)
I
Boltzmann distribution: Fraction of molecules with translational energy 0 : µ ¶ µ ¶ ( 0 ) 0 0 = exp − = exp − (5.50)
I
Resulting rate constant: max
2
µ
8
¶12
× −0
(5.51)
2
µ
8
¶12
× −0
(5.52)
= ×
or max
I
= ×
Arrhenius activation energy: Applying the definition of the apparent (Arrhenius) activation energy , we find that = 2
ln 1 = 0 + 2
(5.53)
As we see, is not constant but (slightly) -dependent: The slope of a plot of ln max −1 −1 vs. increases as → ∞ (or → 0). The Arrhenius preexponential factor would not be constant as well, it is also slightly -dependent.
36
We use and 0 to indicate the energies in molecular units, and and 0 for the respective values in molar units.
5.1 Hard sphere collision theory
100
5.1.3 Allowance for steric hindrance I
The sterical factor p: In addition, even if 0 , not every collision will lead to a reaction, because real molecules are not structureless: A reaction requires a specific orientation of the molecules. This is taken into account by introducing a steric factor : µ ¶12 8 2 max = × × × −0 (5.54)
5.2 Kinetic gas theory
101
5.2 Kinetic gas theory 5.2.1 The Maxwell-Boltzmann speed distributions of gas molecules in 1D and 3D
I
Figure 5.2: Experimental measurement of molecular speed distribution using a rotating disk velocity selector and an effusive molecular beam (orrifice diameter ¿ ).
I
Rotating disk velocity selector: • Flight time of molecules with velocity between two rotating disks (distance ): 1 =
(5.55)
• Time delay of second slit with respect to first slit ( = angle, = angular velocity):
(5.56)
1 = 2
(5.57)
2 = • Transmission only if y
= ×
(5.58)
In practice, one uses a number of velocity selectors in series. A measurement of the 1D ( ) using this method gives speed distribution ( ) =
5.2 Kinetic gas theory
I
102
The 1D Maxwell-Boltzmann speed distribution (Fig. 5.3): ( ) =
r
µ ¶ 2 × exp − 2 2
Note that this distribution is already normalized for
( ) = 1 We check this
−∞
by substituting =
Z∞
(5.59)
r
y = 2
r
y = 2
r
2
(5.60)
y Z∞
−∞
I
( ) =
r
2
Z∞ r
−∞
2 −2 1 = √
Z∞
−∞
Figure 5.3: 1D Maxwell-Boltzmann speed distribution.
1 √ 2 − = √ = 1 (5.61)
5.2 Kinetic gas theory
I
103
Velocity distribution in 3 dimensions: Since the velocities in direction are independent, the probability that a molecule has a velocity between and + , and + , and and + is given be the product of ( ) , ( ) , and ( ) : ( ) = ( ) ( ) ( ) Ã ¡ 2 ¢! ¶32 µ 2 2 + + (5.62) = × exp − 2 2 We are, however, not interested in the distribution of velocities pointing into a specific direction ( ), but in the probability that a molecule has a certain speed || in any direction, i.e., that the velocity vector ends somewhere on a sphere with radius q || = 2 + 2 + 2 (5.63) To obtain this distribution, we must integrate over all polar angles and . This corresponds to replacing the volume element with the volume 42 of a spherical shell with radius and thickness y
I
The 3D Maxwell-Boltzmann speed distribution (Fig. 5.4): () =
µ
2
¶32
¶ µ 2 × 4 × exp − 2 2
(5.64)
Z∞ Note that this distribution is also normalized for () = 1 We check this by 0
substituting
r
= y = 2 2 2 2 =
r
y = 2
µ
2
¶12
(5.65) (5.66)
y µ Z∞ () = 0
2
¶32
× 4
Z∞ 0
2 2 −2
µ
2
¶12
(5.67)
µ ¶32 Z∞ µ ¶32 1 1 1√ 2 −2 = 4 × = 4 × × = 1 (5.68) 4 0
5.2 Kinetic gas theory
104
I
Figure 5.4: 3D Maxwell-Boltzmann speed distribution.
I
Notice the differences between the following quantities: • Value at maximum of (): max = • Mean speed: = • Root mean square value:
I
r
r
2
8
³ ´12 r 3 2 =
Table 5.1: Mean molecular speed values for some gases @ = 298 K gas H H2 He O2 CO2 Hg 238 U
[ m s] 2511 1776 1255 444 379 178 163
(5.69)
(5.70)
(5.71)
5.2 Kinetic gas theory
I
105
Kinetic energy distribution function: We may substitute for in Eq. 5.64: 1 2 2 ¶12 µ 2 y = ¶12 µ ¶12 µ 2 1 1 −12 = × × ( ) = ( )−12 y 2 2 ¶12 µ 1 ( )−12 y = 2
(5.72)
=
(5.73) (5.74) (5.75)
Insertion into Eq. 5.64 gives (see Fig. 5.5) ( ) =
µ
2
¶32
× 4
µ
y ( ) = 2
I
2
µ
¶
−
×
1
¶32
µ
1 × 2
¶12 µ
1
¶12
(5.76)
( )12 −
Figure 5.5: Maxwell-Boltzmann distribution function for the kinetic energy.
The fraction of molecules with 0 is temperature dependent.
(5.77)
5.2 Kinetic gas theory
106
5.2.2 The rate of wall collisions in 3D Earlier, we calculated the rate of wall collisions of the molecules of a gas with mass and number density in a container of volume and the resulting gas pressurer using 1 8 a simplified model by assuming that of all molecules have a fixed speed = 6 in + direction (cf. Fig. 5.6). In this simple picture, we neglected the 3D molecular velocity distribution. We obtained the correct result only due to a cancelation of two 1 errors (assumption of fixed and 3D distribution). In the following, 6 (1) we briefly review the simplified picture, (2) we then proceed to derive the correct results by appropriate averaging over the velocity/speed distribution.
I
Figure 5.6: Simplified model for the calculation of the pressure of an ideal gas.
I
Simplified model for the calculation of the pressure of an ideal gas: • In the time , all gas molecules (mass ) in the volume + hit the wall area . • Number of wall collisions in time (with number density ): = ×
1 × × 6
(5.78)
5.2 Kinetic gas theory
107
• Momentum change due to wall collision, force on wall, and pressure: 1 = 2 × 6 • Gas pressure:
=
y
1 1 1 = = × 2 × 6 1 = 2 3
• Result for 1 mol ( =
(5.80) (5.81)
2 with = 6022 1 × 1023 mol−1 and kin = ): 2
1 2 2 3 = 2 = kin = × = 3 3 3 2 I
(5.79)
(5.82)
Correct derivation of the gas kinetic wall collision frequency: • p Number of molecules with velocities in a volume of with = 2 + 2 + 2 (cf. Fig. 5.7) that will hit the area in the time : ( ) = | |
( ) ( ) ( )
(5.83)
• To find the number of all molecules hitting in , we have to integrate over all positive and over all (positive and negative) and , using the 1D-Maxwell ( ) Boltzmann distributions ( ) = , etc.: ¶32 µ × = 2 µ ¶ µ ¶ µ ¶ Z∞ Z∞ Z∞ 2 2 2 | | exp − exp − exp − 2 2 2 0 −∞−∞
(5.84)
We know that the distributions over and are normalized to 1. Hence, we only have to solve the integral over , which we do by the substitution µ µ µ ¶12 ¶12 ¶12 2 = y = y = (5.85) 2 2 y Z∞ 0
µ µ ¶ ¶12 ¶12 Z∞ µ 2 2 2 −2 | | exp − || (5.86) = 2 0
2 =
Z∞ 2 1 2 = − = 2 0
(5.87)
5.2 Kinetic gas theory
• Result:37 = y
108
µ
2 wall
¶12
= × ×
=× =
µ
2
µ
2
¶12
¶12
This can be rewritten using the mean molecular speed , because µ ¶12 µ ¶12 1 16 1 8 =× wall = × 4 2 4 With =
µ
8
¶12
(5.88)
(5.89)
(5.90)
(5.91)
we thus obtain the • Final result for the gas kinetic collision frequency: wall =
I
1 4
(5.92)
Figure 5.7: On the derivation of the gas kinetic wall collision frequency.
37
µ
Notice that two of the three 2 µ integrations over and , only one
¶12
factors in Eq. 5.84 above are compensated by the ¶12 factor remains. 2
5.2 Kinetic gas theory
I
109
Correct derivation of the gas pressure and the ideal gas equation: • Likewise, we take the differential number of wall collisions for specific speeds (Eq. 5.83) to compute the momentum change ( ) in time ( ) = 2 | | × | | × ( ) ( ) ( ) (5.93) • Inserting again the 1D-Maxwell-Boltzmann distributions and integrating over all positive and all , we find the total momentum change in time : ¶32 × 2× = 2 ¶ µ ¶ µ ¶ µ Z∞ Z∞ Z∞ 2 2 2 2 exp − exp − exp − 2 2 2 µ
0 −∞−∞
(5.94)
again using the substitution = • Pressure for = = =
µ
y =
2
¶12
µ
2
¶12
µ ¶ Z∞ 2 2 × 2 exp − 2
(5.95)
0
µ ¶12 Z∞ 1 2 2 2 2 − =
(5.96)
0
y =
(5.97)
Thus, we have fixed our earlier simple derivations by appropriately integrating over the Maxwell-Boltzmann velocity distribution of the molecules.
5.3 Transport processes in gases
110
5.3 Transport processes in gases 5.3.1 The general transport equation I
→ − Definition of the flux J of a quantity: We consider a generic transport property Γ
(5.98)
which differs in magnitude along the direction, i.e., has the gradient grad Γ
(5.99)
The flux of Γ through an area in time is defined as
and can be written as
− → Γ Γ=
(5.100)
− → Γ = − grad Γ
(5.101)
Kinetic gas theory allows us to derive a general equation for the proportionality constant .
I
Figure 5.8: Derivation of the general transport equation.
5.3 Transport processes in gases
I
111
The general transport equation: We consider the net flux of Γ through an area at = 0 that is transported by gas molecules coming from an area above at = 0 + and from below from an area above at = 0 − (where is the mean free pathlength). Denoting the respective transport quantity per molecule as Γ, these partial fluxes are given by the number of gas kinetic collisions hitting on ( = 0 ) times the Γ carried by each gas molecule (number density ): 1 (Eq. 5.92), we obtain 4 µ µ ¶ ¶ 1 Γ (5.102) Γ+ = × × Γ0 + 4
• Using the gas kinetic wall collision frequency wall =
and
• Net effect:
µ µ ¶ ¶ 1 Γ Γ− = × × Γ0 − 4 1 Γ = Γ+ − Γ− = × × 2
µ
¶ Γ
(5.103)
(5.104)
• Net flux (with − sign to account for the fact that the flux is directed along the downhill gradient of Γ): − → 1 Γ Γ= = − 2
µ
Γ
¶
(5.105)
with given by (see the derivation of wall above) =
µ
8
¶12
(5.106)
and, for only one type of gas molecules A (i.e., A - A collisons), 1 1 = √ 2 2 ×
(5.107)
• If also varies along (diffusion), Eq. 5.105 may be recast by defining the overall 0 transport quantity Γ from the transport quantity per molecule Γ as 0
(5.108)
Γ =Γ into the form − → 1 Γ = − 2
Ã
0
Γ
!
(5.109)
In the following, we shall´ apply Eqs. 5.105 or 5.109 to determine the self-diffusion coef³ ¡ ¢ ¡ ¢ 0 ficient Γ = 1 Γ = , the heat conductivity Γ = , and the viscosity Γ = of gases.
5.3 Transport processes in gases
112
5.3.2 Diffusion I
Figure 5.9: The diffusion in a mixture A + B of one sort of molecules A against a time-independent concentration gradient is described by Fick’s first law.
• Fick’s first law:
µ
− → = = −
¶
(5.110)
• In order to derive an expression for the self-diffusion coefficient (diffusion of an isotope of A in normal A), we make the ansatz 0
(5.111)
Γ = 1 Γ = y
− → 1 = − 2
• Result:
=
µ
¶
1 2
(5.112)
(5.113)
• Fick’s second law (time-dependent concentration gradient): 2 = 2
(5.114)
• Einstein’s diffusion law in 1D:
∆ 2 = 2
(5.115)
• Einstein’s diffusion law in 3D:
∆2 = 6
(5.116)
5.3 Transport processes in gases
113
5.3.3 Heat conduction
I
Figure 5.10: Heat conduction is described by Fourier’s law.
• Fourier’s law (for constant temperature gradient):
• Ansatz: y
• Result:
µ ¶ − → = = −
(5.117)
Γ = = internal energy per molecule
(5.118)
µ ¶ − → 1 = − 2 µ ¶µ ¶ 1 = − 2 ¶µ ¶ µ 1 = − 2 µ ¶ 1 = − 2 1 = 2
with = number density and = specific heat per molecule.
(5.119) (5.120) (5.121) (5.122)
(5.123)
5.3 Transport processes in gases
114
5.3.4 Viscosity
I
Figure 5.11: Newton’s law describes the inner friction in laminar flow.
• Newton’s law:
= −
µ
¶
(5.124)
• Ansatz: Γ = = momentum per molecule in direction y
• Result:
(5.125)
µ ¶ − → 1 ( ) = − 2 1 1 ( ) = = = 1 = 2
with = number density and = mass of molecule.
(5.126)
5.4 Advanced collision theory
115
5.4 Advanced collision theory The hard sphere model is a very primitive model. In reality, we have to account for the following: • The reaction probability depends on exact details of the collision, i.e., hard sphere model → ( )
(5.127)
• Molecules are not hard spheres with = 2 but “soft”, i.e., the reactive cross section depends on the relative speed (or kinetic energy) → () → ( ) • The speed distribution follows the Maxwell-Boltzmann distribution, i.e., → () → ( )
• Molecules in different quantum states behave differently, i.e., → ( ; )
I
(5.128)
(5.129)
(5.130)
Examples: The complexity of the problem is illustrated in Fig. 5.12 by two examples: D( ) + H2 (2 2 2 ) −→ HD( ) + H( )
(5.131)
OH( Ω ) + H2 (2 2 2 ) → H2 O(2 1 2 3 ) + H( ) (5.132)
As can be seen, the reaction rate is expected to depend on (speed, translational energy), (vibrational quantum numbers ( 1 = symmetric stretch, 2 = bend, 3 = antisymmetric stretch)), (rotational quantum numbers), Ω (fine structure state, electronic state). In addition, the products are scattered into the spatial angles . I
Figure 5.12: Quantum state-specific elementary reactions.
5.4 Advanced collision theory
116
We shall now derive the fundamental equation of collision theory that allows us to account for those effects.
5.4.1 Fundamental equation for reactive scattering in crossed molecular beams • Consider the following reactive scattering process (Fig. 5.13):
• Intensity of beam of A:
A+B→C
(5.133)
= ×
(5.134)
Is this reasonable? ⇒ Look at the physical dimensions of × : ¤ ¤ £ £ [ cm s] × molecules/ cm3 = molecules cm2 s = [intensity]
(5.135)
• Depletion of intensity of A due to scattering by B:
= − () × × ×
(5.136)
= −()
(5.137)
↑
y rate equation:
↑
↑
• Rate coefficient for a given value of : () = ()
(5.138)
• Averaging over the speed distribution () leads to the fundamental equation of all improved collision theories: ( ) = h ()i
(5.139)
i.e., ( ) =
R∞
() ( )
(5.140)
0
• Frequently used approximation: Since the explicit form of () is usually not known, we write ( ) = hi h()i (5.141) with hi =
µ
8
¶12
(5.142)
and h()i = -averaged (reactive) cross section
(5.143)
5.4 Advanced collision theory
I
117
Figure 5.13: Derivation of the fundamental equation.
5.4.2 Crossed molecular beam experiments I
Differential and integral reactive cross sections: Experiments with crossed molecular beams allow us to measure differential and integral reactive cross sections (Herschbach 1987, Lee 1987). We consider the reaction ( ) + ( ) −→ ( ) + ( )
(5.144)
where denotes the speeds of the molecules and stands for the quantum numbers of their energy states (electronic, vibrational, rotational, nuclear spin). In an ideal experiment (which has yet to be performed), all of the reactants are defined by, e.g., a specific excitation using laser, and the of the products are determined by some probe laser. The can be determined in a molecular beam experiment using a rotating disk velocity selector (see Fig. 5.14).
5.4 Advanced collision theory
118
I
Figure 5.14: Measurement of differential cross sections in crossed molecular beam experiments.
I
Differential and integral cross sections from crossed molecular beam experiments: • We define the differential cross section ( ) for scattering of products C into the solid angle element Ω at such that ( ) Ω = ( ) × × × ( ) × × ( ) × Ω (5.145) • Corresponding integral cross sections are obtain by integration over and . • Depending on the scattering process of interest, we distinguish between — elastic cross sections ( ): measure of the size of the molecules, — inelastic cross sections ( ): measure of the efficiency of energy transfer processes, — reactive cross sections ( ): measure of reactivity.
5.4 Advanced collision theory
I
119
Figure 5.15: Measurement of differential cross sections in crossed molecular beam experiments.
5.4 Advanced collision theory
120
5.4.3 From differential cross sections σ (. . .) to reaction rate constants k (T ) I
Figure 5.16: The strategy.
I
1 → 2 : Integration over angles: 0
( ) =
Z2 Z 0
I
0
( ) ( ) sin
(5.146)
0
2 → 3 : Summation over all product quantum states α0 : X 0 ( ) = ( )
(5.147)
3 → 4 : Thermal averaging over initial quantum states α: X ( ) = × ( )
(5.148)
0
I
with the Boltzmann factor and the partition function − − = P = − X = −
(5.149) (5.150)
y
( ) =
X
( ) ×
−
(5.151)
5.4 Advanced collision theory
I
121
5 → 6 : Thermal averaging over all ² : X × ( ) ( ) =
X −
=
Z
=
× ( )
− × ( )
I
3 → 5 and 4 → 6 : Laplace Transformation: ( ) ∝
Z∞
× ( ) × −
(5.152)
0
This expression follows from the fundamental equation (Eq. 5.140) by substituting for . The exact transformation (giving the correct factors before the integral) will be carried out below (see Eq. 5.162).
5.4.4 Laplace transforms in chemistry Formally, the transformation (Eq. 5.152) is a Laplace transformation (see Appendix E). A Laplace transform generally described the transformation from a microscopic (“microcanonical”) quantity such as ( ) to a macroscopic (“macrocanonical”, i.e., dependent) quantity such as ( ). I
Definition of the Laplace Transformation: The Laplace transform L [ ()] of a function () is defined as the integral Z∞ () = L [ ()] = () −
(5.153)
0
I
Inverse Laplace Transformation: 1 () = L−1 [ ()] = 2
+∞ Z
()
−∞
with an arbitrary real number .
(5.154)
5.4 Advanced collision theory
I
122
Notes: (1) The so far best known Laplace transform to us is the partition function Z ∞ X − ( ) = = () − (5.155)
0
(2) While the pathway from ( ) to ( ) by forward Laplace transformation is straightforward, it is not generally straightforward to derive ( ) by inverse Laplace transformation from ( ), because ( ) would be needed from = 0 to = ∞.
5.4 Advanced collision theory
123
5.4.5 Bimolecular rate constants from collision theory I
Transformation from velocity to energy space: Taking our results from kinetic gas theory, we are now in the position to apply the fundamental equation of collision theory (Eq. 5.140) Z∞ ( ) = () ( ) (5.156) 0
using various functional forms for . Towards these ends, we first transform from to by inserting () from Eq. 5.64 (with the reduced mass instead of ). We thus obtain µ µ ¶32 Z∞ ¶ 2 3 ( ) = 4 () exp − (5.157) 2 2 0
As done above, we substitute for according to 1 2 2 ¶12 µ 2 y = µ ¶12 µ ¶12 µ ¶12 µ ¶12 1 1 1 2 1 = = y 2 2 µ ¶12 µ ¶12 1 1 y = 2
=
(5.158) (5.159) (5.160) (5.161)
and obtain ( ) =
µ
1
¶12 µ
2
¶32 Z∞ ¶ µ ( ) exp − 0
(5.162)
The integration from 0 to ∞ takes into account that a reaction can occur only if 0 . This expression is interpreted as follows: • The function − describes the fraction of molecules with energy . • The function ( ) in the integrand is called the excitation function38 . This can, in general, be a rather complicated (and not necessarily smooth) function (see Fig. 5.17). • The complete integrand ( ) − is called the reaction function39 , because it describes the reactive fraction of the molecules. • The rate constant ( ) corresponds to the area under the reaction function. 38 39
“Anregungsfunktion” “Reaktionsfunktion”
5.4 Advanced collision theory
I
Figure 5.17: Advanced collision theory: The excitation function.
I
Figure 5.18: Advanced collision theory: The reaction function.
124
5.4 Advanced collision theory
I
125
Non-equilibrium energy distribution:* The above expression for ( ) is valid if the reacting molecules are in internal equilibrium, i.e., the energy states are populated according to the Boltzmann distribution. Deviations from the Boltzmann distributions will occur in the case of very fast reactions, if the reaction is faster than the relaxation between the energy states. In this case, ( ) decreases because the mean energy of the reacting molecule decreases. Mean energy of the reacting molecules (see Fig. 5.19): =
X
Z∞ = ()
(5.163)
0
(1) For thermal equilibrium: () = ()
(5.164)
with () being the Boltzmann distribution. (2) Non-equilibrium: () ()
(5.165)
y non-eq. eq.
(5.166)
I
Figure 5.19: Mean energy of the reacting molecules in thermal equilibrium and for a non-eqilibrium situation.
I
Conclusion: The non-equilibrium distribution leads to a reduced rate coefficient.
5.4 Advanced collision theory
126
a) Advanced collision theory for hard spheres (Fig. 5.20)
I
Figure 5.20: The hard sphere model.
(1) Ansatz for the reactive cross section: ( ) =
½
2 0 0 0
(5.167)
(2) Inserting this ansatz into Eq. 5.162:
( ) =
µ
1
¶12 µ
2
µ ¶32 Z∞ ¶ 2 exp −
(5.168)
0
(3) Result for ( ): With Z
we obtain
µ
( ) =
1
=
¶12 µ
2
( − 1) 2 ¶32
2
¶µ ¶¯∞ µ ¯ − − 1 ¯¯ × ( ) × exp − 0 2
y
2
( ) =
µ
8
(5.169)
¶12 µ ¶ µ ¶ 0 0 1+ exp −
(5.170)
(5.171)
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127
(4) Arrhenius activation energy: = 2
y
ln
µ ¶ 1 0 0 ln ∝ ln + ln 1 + − 2 ¶ µ ¶ µ 0 11 1 0 ln + = + − 2 1 + 0 2 2 0 1 = 0 + − 2 1 + 0
(5.172) (5.173) (5.174)
(5.175)
b) Line-of-Centers model (Figs. 5.21 - 5.22) Even for hard spheres, because of angular momentum conservation (see below), not the entire collision energy, but only the fraction “directed” along the line of centers () is effective. In particular, we have • for a collision with = 0:
=
(5.176)
y central collisions are most effective, • for a collision with = :
= 0
(5.177)
y glancing collisions are ineffective because = 0, • general expression (see below):
µ ¶ 2 = 1 − 2
y decreases with increasing .
(5.178)
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128
Figure 5.21: Collision energy along line-of-centers
Functional expression for ( ): (1) We assume that a reaction can occur only if ≥ 0 . Thus, we first determine (see Fig. 5.21): = cos 2 = 2 cos2 ¡ ¢ = 2 1 − sin2 µ ¶ 2 2 = 1− 2 y
µ ¶ 2 = 1 − 2
(5.179) (5.180) (5.181) (5.182)
(5.183)
(2) We see from this expression that decreases with increasing . Thus, for a given , we have a maximal value of (i.e., a value max ), for which ≥ 0 . We determine max from the condition that µ ¶ 2 = 1 − 2 ≥ 0 (5.184) y
µ ¶ 0 = 2max ≤ 1− µ ¶ 0 2 2 ( ) = max = 1 − 2
y
2
(5.185)
(5.186)
5.4 Advanced collision theory
129
y
⎧ ⎨
µ ¶ 0 0 1 − ( ) = ⎩ 0 0 2
(5.187)
( ) is shown as function of in Fig. 5.22. Crossed molecular beam experiments have indeed shown similar dependencies for a number of reactions. I
Figure 5.22: Reactive cross section for the line-of-centers model.
(3) Result for ( ) from Eq. 5.162: ¶32 µ ¶12 µ 2 1 ( ) = ∞ µ ¶ ¶ µ Z 0 2 exp − × 1 − 0
¶12 µ ¶32 2 1 = µ ¶ Z∞ 2 × ( − 0 ) exp − µ
(5.188)
(5.189)
0
y
( ) =
µ
8
¶12
¶ µ 0 exp − 2
(5.190)
This result is identical to that of our primitive “hard sphere-fixed mean relative speed” model from Section 5.1!
5.4 Advanced collision theory
130
(4) Arrhenius activation energy: ln
(5.191)
1 = 0 + 2
(5.192)
= 2 y
c) Allowance for the “steric effect” In order to account for the “structures” of the molecules which can undergo a reaction only if the reaction partners have a specific relative orientation, we introduce a (generally -dependent) steric factor ( ): µ
8 ( ) = ( )
¶12
¶ µ 0 exp − 2
(5.193)
In ( ), we include all sorts of factors leading to deviations from the predicitions by simple collision theory. In favorable cases, ( ) can be predicted by theory. Towards these ends, we define a reaction probability ( ) (also called the opacity function, and calculated by theory) and integrate according to
( ) =
Z0max
( ) 2
(5.194)
0
In other cases, it may be convenient to define an angle dependent reaction threshold energy ∗0 : (5.195) ∗0 = 0 + 0 (1 − cos ) y
∗0 = 0 for = 0 ∗0 0 for 6= 0
(5.196)
Such models are helpful for explaining steric factors in the range of 1 ≤ ≤ 0001.
5.4 Advanced collision theory
I
Figure 5.23: Collision theory: Allowance for steric effects.
I
Table 5.2: Comparison of predictions by collision theory with experimental data.
131
5.4 Advanced collision theory
I
132
Harpooning: The value of ≥ 1 for the steric factor of the K + Br2 reaction can be understood by the “harpooning mechanism”: An electron jumps from the K to the Br2 at a long intermolecular distance. The subsequent collision and reaction at closer distance then occurs between a K+ and Br− 2 . The collision between these two is strongly affected by the Coulomb attraction.
5.4.6 Long-range intermolecular interactions To this point, we have neglected any interactions between the reacting molecules other than the the collision itself. Thus, only straight line trajectories were allowed. In reality, however, there will be attractive or repulsive interactions between the colliding molecules (see Fig. 5.24). These will inevitably affect the collision dynamics. An extreme example is the K + Br2 reaction mentioned above. We will investigate the potential energy hypersurfaces governing the reactions in Section 6. Here, we will consider only long-range intermolecular interactions which can be easily implemented in collision theory. I
Figure 5.24: Examples of attractive and repulsive trajectories for the collision between two molecules.
a) Types of long-range interactions The long-range interactions between molecules can be described using multipole expansions of the potentials (charge: 1; dipole: 2; induced dipole: 3; quadrupole: 3 . . . In general, the interaction potential between two multipoles of ranks and 0 is proportional to − with = + 0 − 1: µ ¶ () = − (5.197)
5.4 Advanced collision theory
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133
Table 5.3: Types of long-range intermolecular interactions. system
()
1 2 2 charge-charge (Coulomb) () = 40 → − cos → charge-dipole ( ) = − · =− 40 2 3 (μ1 · r) (μ2 · r) μ1 · μ2 − 2 dipole-dipole ( 1 2 ) = 3 40 charge-quadrupole Θ ∝ −3 dipole-quadrupole Θ ∝ −4 charge-induced dipole ∝ −4 quadrupole-quadrupole ΘΘ ∝ −5 induced dipole-induced dipole ∝ −6
I
3 Table 5.4: Long-range intermolecular interactions ( = 15 D, = 3 ˚ A , = 1, = 0; energies in kJ mol; for comparison: = 25 kJ mol @ = 298 K). ³ ´ ³ ´ ˚ ˚ system = 5 A = 10 A ion-ion 278 139 ion-dipole 17.4 4.35 ion-induced dipole 3.34 0.21 dipole-dipole 2.17 0.27
b) Lennard-Jones (6-12) potential for nonpolar molecules A realistic intermolecular potential for non-polar molecules is the Lennard-Jones (6-12) potential ∙³ ¸ LJ ´12 ³ LJ ´6 LJ () = 4LJ − (5.198) LJ is usually given in reduced form in units of K as ∗LJ =
LJ
(5.199)
5.4 Advanced collision theory
134
I
Figure 5.25: Schematic plot of the Lennard-Jones potential.
I
Estimation of LJ parameters: Values for LJ and LJ are typically derived from transport properties (e.g., viscosity), thermodynamic data (vdW parameters), or — if there is no better way — from empirical correlations (see, e.g., Svehla1963) with critical data, boiling temperatures, . . .
I
Table 5.5: Lennard-Jones for some gases (Mourits1977). He Ne Ar Kr Xe H2
LJ ( pm) (LJ ) ( K) 2560 102 2822 320 3465 1135 3662 1780 4050 2302 2930 370
LJ ( pm) (LJ ) ( K) N2 3738 820 O2 3480 1026 CH4 3790 1421 CF4 4486 1673 CCl4 5611 4155 CO2 3943 2009 SF6 5199 2120
5.4 Advanced collision theory
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135
Figure 5.26: Lennard-Jones interaction potentials for He, Ne, Ar, Kr, and Xe.
c) Orbital angular momentum and centrifugal barrier* In the case of a non-central collision, part of the translational energy is converted to “rotational energy” (orbital angular momentum; see Fig. 5.27). This is seen by partitioning the velocity vector between the direction ( ) and the radial direction ( ). The component corresponds to a rotational motion. I
Figure 5.27: Orbital angular momentum in binary collisions.
5.4 Advanced collision theory
136
• Radial velocity component: cos =
=
y = • Kinetic energy:
1 2 |2 {z }
=
+
translation via LC
• Angular velocity: =
(5.200)
(5.201) 1 2 |2 {z }
(5.202)
1 = 2 2 (rotation) 2
= = = = 2
(5.203)
• Kinetic energy: 1 1 ( )2 + ( )2 2 2 1 · 2 1 2 2 = + 2 2 2 2 1 2 1 · = + 2 2 2 1 · 2 1 2 2 2 = + 2 2 2
(5.204)
=
(5.205) (5.206) (5.207)
• Orbital angular momentum: is the orbital angular momentum = 2 y
= =
(5.208)
1 ·2 2 = + 2 22
(5.209)
From quantum mechanics, 2 = ~2 ( + 1) with as orbital angular momentum quantum number. y
where 2 =
1 · 2 ~2 ( + 1) = + 2 22 1 ·2 = + 2 ( + 1) 2
(5.210) (5.211)
~2 is formally an -dependent rotational constant. 22
• Centrifugal barrier: The centrifugal kinetic energy term
~2 ( + 1) acts as a 22
centrifugal barrier
1 2 ~2 ( + 1) centrifugal = 2 = 2 22
(5.212)
5.4 Advanced collision theory
137
¡ ¢ • With a long-range attractive term of the form − , the intermolecular potential thus has the form µ ¶ 2 () = − + 2 (5.213) µ ¶ ~2 ( + 1) = − + (5.214) 22 i.e., (5.215) () ∝ −− + −2 — The −2 term is important in practice if 3. In effect, it leads to an effective energy barrier (“centrifigual” barrier). — The larger for a given , the larger the orbital angular momentum . I
Figure 5.28: Potential energy curves with centrifugal barriers for a diatomic system.
d) Langevin “capture” rate constant for ion-molecule reactions* An nice illustration of the above results is the rate constant for ion-molecule reactions. The intermolecular interaction is governed by the point charge-induced dipole interaction.40 In addition, the centrifugal barrier has to be taken into account, so that µ ¶2 ()2 () = − + (5.216) 2 4 (40 ) 2 The resulting expression for the rate constant is !12 Ã µ ¶ 2 4 () 1 ×Γ ( ) = 2 2 (40 ) Typical values of are 100 × 40
() needs to be checked again!!
(5.217)
5.4 Advanced collision theory
138
6. Potential energy surfaces for chemical reactions 6.1 Diatomic molecules I
Morse potential (anharmonic oscillator): () = [1 − exp (− ( − ))]2
I
I
I
Vibrational states of a harmonic oscillator: µ µ ¶ ¶ 1 1 = + = ~ + 2 2 Vibrational states of an anharmonic oscillator: µ µ ¶ ¶2 1 1 − ~ + + = ~ + 2 2
(6.2)
(6.3)
Dissociation energies D and D0 : 1 1 = 0 + ~ − ~ + 2 4
I
(6.1)
Figure 6.1: Potential energy curve of diatomic molecules.
(6.4)
6.2 Polyatomic molecules
139
6.2 Polyatomic molecules The potential energy becomes dependent of the different internal coordinates. I
Question: How many internal coordinates (dimensions) do we need?
I
Answer: • Total number of degrees of freedom of an -atomic molecule: 3
(6.5)
• for translational motion of center of gravity: 3
(6.6)
• for rotational motion around center of gravity 2 for linear molecules 3 for nonlinear molecules
(6.7) (6.8)
• resulting number of vibrational coordinates: — linear molecules: 3 − 5
(6.9)
3 − 6
(6.10)
— nonlinear molecules:
I
3N − 6 dimensional potential energy hypersurface: • The internal coordinates are described using the normal coordinates (vibrational displacements). • The potential energy function (1 2 3 −6 )
(6.11)
is a hypersurface in a 3 − 6 (3 − 5) dimensional hyperspace. But we can plot only in 2-D or 3-D space.
6.2 Polyatomic molecules
140
I
Figure 6.2: Potential energy hypersurface for a simple atom transfer reaction (plot for fixed angle, e.g., collinear collision and other fixed coordinates depending on dimensionality of the system).
I
Figure 6.3: Schematic potential energy hypersurface of a triatomic molecule.
6.2 Polyatomic molecules
I
Figure 6.4: Potential energy hypersurface of HCO.
I
Figure 6.5: Potential energy hypersurface of HCO.
141
6.2 Polyatomic molecules
I
142
Models of potential energy surfaces: (1) Empirical or semiempirical models (e.g., LEPS), (2) Single points ( ) by ab initio calculations using quantum chemistry. Problem: Fitting the surface to the computed points.
I
Kinetics and Dynamics: (1) TST = Transition State Theory (statistical theory), (2) Classical trajectory calculations (by solving Newton’s equations), (3) Fully quantum mechanical wave packet calculations (by solving the timedependent Schrödinger equation).
6.3 Trajectory Calculations
143
6.3 Trajectory Calculations The “exact” dynamics of the molecules on a potential energy hypersurface can be followed by classical trajectory calculations. I
Phase Space: 6-dimensional space spanned by the spatial coordinates ( ) and the conjugated momenta ( ): ↔ (6.12) where =
I
· =
Hamilton function:
2 + () 2 Note that depends only on and depends only on . = + =
I
(6.13)
(6.14)
Equations of motion: • for a 1 particle system: (1) () =−
(6.15)
=−
(6.16)
= = y
·
=− since contains only not . (2)
2 + 2 µ ¶ · · 1 1 = × = × = =
y y
·
= • for an atomic system: = + =
3 X =1
2
2 + (1 2 3−6 )
·
=
(6.18) (6.19)
(6.20)
(6.21)
(6.22)
= − ·
(6.17)
Thus we obtain a set of coupled differential equations which can be solved using the Runge-Kutta or Gear methods.
6.3 Trajectory Calculations
144
• Thermal rate constants ( ) follow by suitable averaging over the initial conditions ( vibrational states, rotational states, angles, . . . ). • Major problems: Zero-point energy? Tunneling? Surface crossings? I
Figure 6.6: Trajectory calculations.
I
Figure 6.7: Trajectory calculations.
6.3 Trajectory Calculations
145
I
Figure 6.8: The reaction H + Cl2 is a reaction with an early TS. The reaction is known to yield HCl with an inverted vibrational state distribution (⇒ HCl-Laser).
I
Figure 6.9: Bimolecular reaction with a late TS.
6.3 Trajectory Calculations
146
7. Transition state theory 7.1 Foundations of transition state theory The concept of transition state theory (TST) or activated complex theory (ACT) goes back to H. Eyring and M. Polanyi (1935). TST is a statistical theory; it does not give information on the exact dynamics of a reaction. It does, however, give (statistical) information on product state distributions. There are two forms of TST: • Macrocanonical TST ⇒ ( ) for a macrocanonical ensemble of molecules with a constant (an ensemble of molecules with energy states populated according to the Boltzmann distribution - each state being equally likely.) • Microcanonical TST (TST) ⇒ () for a microcanonical ensemble of molecules with specific energy . We will consider only the first version. The detailed application of TST also depends on whether or not the reaction has a distinctive potential energy barrier (⇒ conventional TST or variational TST).
I
Figure 7.1: Transition State Theory (TST): Fundamental equations and applications.
7.1 Foundations of transition state theory
147
7.1.1 Basic assumptions of TST (1) The Born-Oppenheimer approximation (separation of electronic and nuclear motion) is a valid approximation. Thus we can describe the nuclear motion during the reaction on a single electronic potential energy surface (PES; see Fig. 7.2).41 (2) The reactant molecules are in internal equilibrium (the quantum states are populated according to the Boltzmann distribution). (3) “Transition state” and “dividing surface“: a) Conventional TST (valid for so-called “type I” PES’s = PES’s with a distinctive energy barrier along the reaction coordinate (RC)): We can localize a distinctive “transition state” (TS) for the reaction at the saddle point on the PES that separates the reactant and product “valleys”. The TS is located at the maximum of the PES along the minimum energy path (MEP; see Fig. 7.3). b) Variational TST (version that can be used for so-called “type II” PES’s = PES’s without a clear energy barrier along the RC (we can’t localize a simple TS)): We can localize a 3 − 7-dimensional “dividing surface” (DS) on the 3 − 6-dimensional PES that separates the reactant and product regions on the PES in such a way that the “reactive flux” through the DS becomes minimal (Fig. 7.2). The TS is then placed at the point where the MEP intersects the DS. In this case, the position of the TS depends strongly on the finer details such as angular momentum (because of centrifugal barriers which move inwards with increasing or increasing ). (4) The motion along the minimum energy path across the TS (through the DS) can be separated from other degrees of freedom and treated as translational motion. At the TS, we therefore have 1 translational degree of freedom ‡ (the RC) and 3 − 7 (vibrational) degrees of freedom orthogonal to the RC. (5) Reactant molecules passing through the DS never come back (no recrossing). (6) The states of the TS are in quasi-equilibrium with the states of the reactant molecule even if we have no equilibrium between reactants and products (the individual molecules don’t know that this equilibrium is not present - this quasiequilibrium assumption can be dropped in dynamical derivations of TST; see below). I
Conditions for TS at the saddle point: (1) has maximum along ‡ : 2 = 0 and 0 ‡ ‡2
(7.1)
y 1 imaginary frequency (criterion for TS in quantum chemical calculations)! 41
TST cannot be used at least in the simple form considered in this lecture when two potential surfaces are involved that are energetically close to each other (or even become degenerate) and the reaction involves a “non-adiabatic transition” between these two potentials.
7.1 Foundations of transition state theory
148
(2) has minimum along all 3 − 7 other coordinates: 2 = 0 and 0 2
(7.2)
y 3 − 7 real other frequencies. I
Figure 7.2: Transition state theory: The dividing surface with minimal reactive flux.
I
Figure 7.3: Minimum energy path (MEP) and transition state (TS) of a simple atom transfer reaction.
7.1 Foundations of transition state theory
149
7.1.2 Formal kinetic model In order to derive the TST expression for ( ), we first look at a simple formal kinetic model. This formal kinetic model has its roots in the original idea that the TS can be considered as some sort of an “activated complex” with a local energy minimum on the PES. It is not a good derivation of the TST expression for ( ), but leads to the correct expression. I
Activated complex model of TST: 1
3
A + BC À ABC‡ → P
(7.3)
2
y
I
¤ £ 1 [A] [BC] ABC‡ = 2 + 3
Assumption: 3 ¿ 2 y £ ¤ 1 ABC‡ = [A] [BC] = 12 [A] [BC] 2
(7.4)
(7.5)
y Equilibrium between the reactants (A + BC) and the TS (ABC‡ ): £ ¤ [P] = 3 ABC‡ = 3 12 [A] [BC] | {z }
(7.6) (7.7)
=
y I
ACT expression for rate constant: = 3 12 Below, we will find that 3 = ‡ = and 12
¶ µ ∆‡ ∝ exp −
(7.8)
(7.9) (7.10)
7.1.3 The fundamental equation for k (T ) The above derivation of the TST rate constant was based on an oversimplified picture, since the TS is not at a local minimum but at a saddle point on the PES. In this section, we derive the TST fundamental equation more carefully by bringing in some dynamical aspects. In (1) - (4), we explore the implications of the quasi-equilibrium assumption, and in (5) - (9) we develop a dynamical semiclassical model:42 42
Semiclassical because the translation is handled classically while all other degrees of freedom are described quantum mechanically.
7.1 Foundations of transition state theory
150
(1) We start by assuming equilibrium between the reactants and products: 1
3
2
4
A + BC À ABC‡ À P
(7.11)
(2) We consider two dividing surfaces at the TS separated by a distance of ≤ .
is the “length” of a “phase space cell” in semiclassical theory which is determined by Heisenberg’s uncertainty principle: ∆ ‡ × ∆‡ = × ∆‡ =
(7.12)
(3) We now consider the number densities of molecules passing through these surfaces − → ← − from left to right ( ‡ ) and from right to left ( ‡ ). The total number density ‡ at the TS in equilibrium is their sum, i.e., − → ← − ‡ = ‡ + ‡ = ‡ A BC In equilibrium, we must have
− → ← − ‡ = ‡
(7.13)
(7.14)
Thus,
− → ← − 1 ‡ ‡ = ‡ = ‡ = A BC (7.15) 2 2 ← − (4) We now suddenly remove all the products. Then ‡ = 0 However, there is no − → reason for ‡ to change, because the molecules on the reactant side remain in internal equilibrium (this is the quasi-equilibrium assumption).43 y − → ‡ ‡ (7.16) = A BC ‡ = 2 2 − → Thus, we can still express ‡ using the equilibrium constant ‡ even if there is no equilibrium between reactants and products. (5) We now turn to the reaction rate, which is given by the number density of mole− → cules ‡ in the time interval that are crossing the DS in the forward direction. We can write this as → − − → →‡ ‡ ‡ ‡ ‡ − = = (7.17) where → − • ‡ is the decay rate of the TS in the forward direction (or the frequency factor for the molecules crossing the DS in the forward direction), 43
−→ The quasi-equilibrium assumption for ‡ is made in canonical TST. In a microcanonical derivation · · −→‡ ←−‡ of the TST expression for (), we consider the fluxes and through the dividing surface at the TS. Then, the individual molecules “dont’t know” that the products have been removed and · · ←−‡ −→‡ = 0. Since they don’t know about this, the flux has to remain unchanged.
7.1 Foundations of transition state theory
151
→ − → − • ‡ = ‡ is the mean speed of the molecules crossing the DS in the forward direction. (6) We now use the equilibrium constant to determine ‡ via ‡ ABC A BC ‡ = ‡ A BC y ‡ = ABC
‡ =
y
with
(7.18) (7.19)
→‡ − → − ‡ = ‡ A BC = A BC
(7.20)
−‡ → = ‡
(7.21)
(7) From statistical thermodynamics (see Appendix G), we have the following expression for the equilibrium constant ‡ ( ) in terms of the molecular partition functions: ¶ µ ∗ ∆0 ‡ ( ) = (7.22) = exp − • The ’s are the respective molecular partition functions (per unit volume). • ∗ is written with respect to the zero-point level of the reactants.
• The exp (−∆0 ) factor arises subsequently, because we now express the value of partition function for the TS relative to the zero-point level of the TS (∗ = exp (−∆0 )). (8) Separating the motion along the reaction coordinate ‡ from the other degrees of freedom, we write = ‡ ‡ (7.23) where ‡ is the 1-D (translational) partition function describing the translational motion of the molecules along ‡ across the TS and ‡ is the partition function for all 3 − 7 other (rotational-vibrational-electronic) degrees of freedom. Here,
• the 1-D translational partition function is ‡ =
µ
2 2
¶12
×
(7.24)
• the mean speed along ‡ for crossing the DS in the forward direction is given by R ∞ −2 2 ¶12 µ →‡ − 0 = R ∞ −2 2 (7.25) = 2 −∞
7.1 Foundations of transition state theory
152
(9) Collecting and inserting the results into −‡ → ( ) = ( ) we obtain 1 ( ) = y
I
µ
2
¶12
×
µ
2 2
¶12
¶ µ ‡ ∆0 × exp −
¶ µ ∆0 ‡ exp − ( ) =
(7.26)
(7.27)
(7.28)
Result: In order to account () for some recrossing of the TS and () for quantum mechanical tunneling, we add an additional factor called the transmission coefficient. For well-behaved reactions, ≈ 1. Thus, we end up with the fundamental equation of TST in the form: ¶ µ ‡ ∆0 ( ) = exp − (7.29) Note that ‡ is the partition function of the TS excluding the RC. Equation 7.29 allows us to calculate ( ) using the following data: • The threshold energy for the reaction ∆0 (or, per mol, ∆0 ). ∆0 has to be obtained from ab initio quantum chemistry calculations, and • structural information on the reactants and the TS (i.e., bond lengths and angles, vibrational frequencies).
I
Figure 7.4: Illustration of ∆ ‡ , ∆0 , and .
7.1 Foundations of transition state theory
153
∆ ‡ : Born-Oppenheimer potential energy barrier ∆0 : zero-point corrected threshold energy for the reaction : Arrhenius activation energy
(7.30) (7.31) (7.32)
∆0 is given with respect to the zero-point energies of the reactants and the TS, =
1X 2
(7.33)
7.1.4 Tolman’s interpretation of the Arrhenius activation energy I
We now understand the difference between E and ∆E0 : ln ( ) (7.34) Ã µ ¶! ‡ ∆0 2 exp − ln (7.35) = µ ¶ ‡ 2 ln 2 ln 2 ln (7.36) − + = ∆0 + + } {z | {z } | =hreacting molecules i =hall molecules i
= 2
• ∆0 is difference of the zero-point levels of the reactants and the TS, • the term is the mean translational energy in the RC (from the factor), ln • the 2 terms are the internal (vibration-rotation) energies of the TS and the reactants, respectively. y
I
Tolman’s interpretation of E (see Fig. 7.4): = hreacting molecules i − hall molecules i
(7.37)
7.2 Applications of transition state theory
154
7.2 Applications of transition state theory ¶ µ ‡ ∆0 ( ) = exp −
(7.38)
TST has gained enormous importance in all areas of physical chemistry because it provides the basis for understanding – and/or calculating – • quantitative values of preexponential factors, • ∆0 from measured values, • -dependence of and deviations from Arrhenius behavior, • gas phase and liquid phase reactions (proton transfer, electron transfer, organic reactions, reactions of ions in solutions), • pressure dependence of reaction rate constants (activation volumes), • kinetic isotope effects, • structure-reactivity relations (e.g., Hammett relations in organic chemistry), • features of biological reactions (enzyme reactions), • heterogeneous reactions (heterogeneous catalysis, electrode kinetics), . . . 7.2.1 The molecular partition functions In order to evaluate Eq. 7.29 for a given reaction, we need, besides ∆0 , the molecular partition functions for the reactants and the TS. We summarize only the main points here; further information is given in Appendix G. I
Definition: =
I
P
−
(7.39)
Physical interpretation: • The molecular partition function is a number which describes how many states are available to the molecule at a given temperature . • The ratio
‡ (7.40) in Eq. 7.29 is thus the ratio of the number of states available to the TS and ‡ the reactant molecules. Since each state is equally likely, is the relative probability of the TS vs. the probability of the reactants.
Note again that ‡ is the partition function of the TS excluding the translational motion along the RC.
7.2 Applications of transition state theory
I
155
Separation of degrees of freedom: For separable degrees of freedom, we can write the molecular partition function as (7.41)
= × × × Nuclear spin ( ) may have to be taken into account in some cases as well.
I
Table 7.1: Molecular partition functions. type of motion
energy 2 2 translation (1-D) 8 2 ¶ µ 2 2 2 2 translation (3-D) + + 8 2 2 2 ~2 2 rotation (1-D)a 2 ~2 ( + 1) rotation (2-D)b 2
a
b c d
rotation (3-D)c
or
vibrationd electronic
partition function µ ¶12 2 2 µ ¶32 2 2 µ ¶12 2 2 µ ~ ¶ 2 ~2 ¶32 µ 12 2 ( )12 2 ~ [1 − exp (− )]−1 P exp (− )
magnitude 108 × 1024 × 10 102 103 104 1 10 1
Free internal rotor. For a hindered rotor (torsional vibrational mode), the partition function has to be determined by an explicit calculation over the hindered rotor quantum states. Linear molecule. Nonlinear molecule. For each harmonic oscillator degree of freedom measured from zero-point level.
7.2 Applications of transition state theory
156
7.2.2 Example 1: Reactions of two atoms A + B → A· · · B‡ → AB This example is somewhat artifical unless there is some mechanism to remove the bond energy (for example, in associative ionization, A + B → A· · · B‡ → AB+ + − ). ¶ µ ∆0 ‡ (7.42) exp − ( ) = Ã Ã ! ! µ ¶ ‡ ‡ ∆0 = exp − (7.43) ( ) ( ) 1 ¶ µ ¶32 µ 2 ‡ 2 8 2 2 ( + ) = 2 2 2 2 ¶ µ ∆0 (7.44) × exp − ! ¶32 Ã 2 µ 8 ( + )2 2 + = 2 2 + µ ¶ ∆0 × exp − (7.45)
y ( ) =
µ
8
¶12
¶ µ ∆0 ( + ) exp − 2
This result is identical to that obtained by the LC model in collision theory!
(7.46)
7.2 Applications of transition state theory
157
7.2.3 Example 2: The reaction F + H2 → F· · · H· · · H‡ → FH + H I
Exercise 7.1: Evaluate the rate constant for the reaction F + H2 → F · · · H · · · H‡ → FH + H
(7.47)
using the data in Table 7.2 at = 200, 300, 500, 1000 and 2000 K, determine the Arrhenius parameters, and compare the results with the experimental value of = 2 × 1014 exp (−800 ) cm3 mol s
(7.48) ¤
I
Table 7.2: Properties of the reactants and transition state of the reaction F + H2 .
a b c d
I
parameter 2 (F · · · H) ( ˚ A) 1 (H · · · H) ( ˚ A) −1 1 ( cm ) 2 ( cm−1 ) 3 ( cm−1 ) 4 ( cm−1 ) ³( u) ´ 2 u ˚ A d ¡ ¢ 0 kJ mol−1
F · · · H · · · H‡ a 1602 0756 40076 3979 3979 3108 c 21014 7433 1 4 657
F
H2 07417 43952
b
189984 2016 0277 2 4 1 000
The transition state FHH‡ is assumed to be linear. = (H-H). One imaginary frequency describes the TS along the RC. The electronic ground state of F is 2 32 . We neglect the 2 12 spin-orbit component at (2 12 ) = 404 cm−1 because it does not correlate with the products H + HF. The electronic state of linear FHH‡ is 2 Π. The ground state of H2 is 2 Σ+ .
Solution 7.1: ¶ µ ∆0 ‡ (7.49) exp − ( ) = 2 Ã ! Ã ! Ã ! Ã ! ‡ ‡ ‡ ‡ = ( ) (2 ) 2 2 2 ¶ µ ∆0 (7.50) × exp −
7.2 Applications of transition state theory
158
The different factors can be evaluated ! Ã ‡ ( ) (2 ) ! Ã ‡ 2 Ã ! ‡ 2
separately: µ
=
Ã
‡ 2
!
¶32 µ
¶32 2 = 2 ¶ µ ‡ 8 2 ‡ 2 = = 2 8 2 2 2 2 2 + 2 2
1 − exp (− 2 ) ³ ´i 1 − exp − ‡
3 h Y
(7.51) (7.52) (7.53)
=1
=
4 4×1
(7.54)
Note: The difference between the TST result and experiment can be ascribed (a) to tunneling and (b) deficiencies of the ab initio PES. Indeed, according to newer ab initio calculations, the TS is actually slightly bent and the TS parameters differ slightly, yielding better agreement with experiment. ¥
7.2 Applications of transition state theory
159
7.2.4 Example 3: Deviations from Arrhenius behavior (T dependence of k (T ))* I
Exercise 7.2: The temperature dependence of bimolecular gas phase reactions of the type A + B → products can usually be expressed in the form ( ) = × × exp (−∆0 )
(7.55)
Using the TST expression for ( ) ¶ µ ∆0 ‡ , exp − ( ) =
(7.56)
determine the values of for the types of reactions in Table 7.3, assuming that the vibrational degrees of freedom are not excited. ¤ I
I
Table 7.3: A
+B
atom
atom
linear TS
atom
linear molecule
linear TS
transl. 32 1 32 32 1 −32
atom
linear molecule
nonlin. TS
1
−32
atom
nonlin. molecule
nonlin. TS
1
−32
linear molecule linear molecule
linear TS
1
−32
linear molecule linear molecule
nonlin. TS
1
−32
linear molecule nonlin. molecule
nonlin. TS
1
−32
nonlin. molecule nonlin. molecule
nonlin. TS
1
−32
→ TS‡
rot. 1 1 1 132 1 32 32 1 132 1 1 1 32 32 1 32 32 32
+05 −05 00 −05 −15 −10 −15 −20 ¥
Solution 7.2: See Table 7.3.
Using the above results, we can recast ( ) and determine the following expression for the Arrhenius activation energy, with as above: = ∆0 + +
‡ 2 ln
X ln reactants − Reactants 2
(7.57)
We see that the value for depends on the vibrational frequencies of the TS and the reactants. Often, the frequencies of the reactants are very high so that reactant = 1. ‡ However, the TS may have soft vibrations so that may approach the classical value for → ∞ (7.58) Thus, each “soft” vibrational mode in the TS increases by 1, each “soft” vibrational mode in the reactants decreases by 1. = [1 − exp (− )]−1 →
7.3 Thermodynamic interpretation of transition state theory
160
7.3 Thermodynamic interpretation of transition state theory 7.3.1 Fundamental equation of thermodynamic transition state theory As shown above,
‡ (7.59) ‡ can often be interpreted using thermodynamic arguments, i.e., ∆‡ , ∆ ‡ , and ∆ ‡ . For liquid phase reactions, this is straightforward. For gas phase reactions, we have to take into account the difference between ‡ and ‡ . ( ) =
a) Thermodynamic transition state theory for unimolecular gas phase reactions For unimolecular isomerization and dissociation reactions of the type ABC → ABC‡ → ACB‡
(7.60)
ABC → ABC‡ → A + BC‡ ,
(7.61)
or we have ‡
¤ £ ABC‡ ‡ ( ) = = ABC = ‡ ( ) [ABC] ABC
(7.62)
‡ ( ) = ‡ ( ) = ‡ ( ) is dimensionless (∆ ‡ = 0) and we do not have to worry about a difference between ‡ ( ) = ‡ ( ) and about units or different standard states. I
Results: Eq. 7.59 thus gives the following results: • Rate constant:
y
µ ¶ ‡ ∆‡ ( ) = ( ) = exp −
(7.63)
µ ¶ µ ¶ ∆ ‡ ∆ ‡ exp exp − ( ) =
(7.64)
• Arrhenius activation energy: = y
∙ µ ¶ µ ¶¸ ∆ ‡ ∆ ‡ = ln exp exp −
2 ln
2
∆ ‡ ‡ ∆ ‡ = ∆ + +
= ∆ ‡ + + 2
(7.65)
(7.66) (7.67)
If ∆ ‡ 6= ( ) in the temperature range of interest, we obtain = ∆ ‡ +
(7.68)
7.3 Thermodynamic interpretation of transition state theory
161
• Arrhenius preexponential factor:
y y
∆ ‡ = −
(7.69)
µ ¶ µ ¶ ∆ ‡ ( ) = exp exp −
(7.70)
µ ¶ ∆ ‡ = exp
(7.71)
b) Thermodynamic transition state theory for bimolecular gas phase reactions I
Conversion from K‡ (T ) to K‡ (T ): For bimolecular reactions A + BC → ABC‡ ,
(7.72)
we have to account for the different numbers of species ∆ ‡ = ‡ − reactants : ∆ ‡ = −1 .
(7.73)
We therefore have to convert from ‡ ( ) to ‡ ( ) via ABC‡ ABC‡ ª ¤ £ ¶−∆ ‡ µ ‡ ABC 1 ‡ ‡ = A . ( ) = BC = A ª BC ª × ª = ( ) × ª [A] [BC] (7.74) −1 44 ‡ ‡ ‡ Here, ( ) is conveniently given in units of l mol . However, , ∆ , ∆ ‡ and ∆ ‡ are related to the standard state at ª = 1 bar .45 To keep track of units in the l bar conversion , it is useful to apply in by writing mol K = 83145
44
45
m3 Pa l bar J = 83145 = 0083145 = 0 mol K mol K mol K
(7.76)
¡ ¢∆ With ‡ ( ) in units of mol l−1 , we get as close as possible to the standard state ∗ = −1 1 mol kg for reactions in aqueous solution. We recall that the thermodynamic equilibrium constant ( ) = exp (−∆ ª ) referred to the standarrd state at ª = 1 bar is dimensionless. Instead of the dimensionless ‡ ( ), one may also decide to use the equilibrium constant in pressure units ‡ 0 : i h ABC‡
ABC‡ ‡0 0 ‡ ( ) = = A BC = ( ) × [A] [BC]
(7.75)
7.3 Thermodynamic interpretation of transition state theory
162
so that, with ∆ ‡ = −1, we obtain ABC‡ ª µ 0 ¶ 1 ‡ ‡ ( ) = . ª ª × ª = ( ) × A BC ª
(7.77)
Thus, via Eq. 7.59: ¶ µ 0 ‡ 0 ∆ª‡ ( ) = , ( ) = exp − ª ª in units of
µ
dim ( ( )) = dim
¶
µ
0 × dim ª
¶
=
l mol s
(7.78)
(7.79)
and with the standard state for the thermodynamics quantities explicitly indicated by the ª symbol. I
Results: • Rate constant:
¶ µ ¶ µ ∆ ª‡ 0 ∆ ª‡ exp − ( ) = exp ª
(7.80)
• Arrhenius activation energy: = 2
ln
(7.81)
If ∆ ª‡ 6= ( ) in the temperature range of interest: = ∆ ª‡ + 2
(7.82)
• Arrhenius preexponential factor: ∆ ª‡ = − 2 y
¶ µ ¶ µ − 2 ∆ ª‡ 0 exp − exp ( ) = ª ¶ µ ¶ µ 0 ª‡ ∆ 2 exp − exp = ª
(7.83)
(7.84) (7.85)
Comparison with the Arrhenius expression = exp (− ) thus gives ¶ µ 0 ∆ ª‡ exp ª
2
=
(7.86)
7.3 Thermodynamic interpretation of transition state theory
163
• Note: and are given above in units of l mol−1 s−1 (we may denote this by writing them as and ), while ∆ ª‡ and ∆ ª‡ used above are referred to the standard state for gases of ª = 1 bar. — For ideal gases, ∆ ‡ and ∆ ‡ are related via ∆ ª‡ = ∆ ª‡ + ∆ ‡ ( ) = ∆ ª‡ − = ∆ +‡ −
(7.87)
— Likewise, ∆ ª‡ and ∆ +‡ for are related via Z + Z + Z + Z + ∆‡ +‡ ª‡ ∆ − ∆ = = = = (7.88) ª ª ª ª + + ª ‡ ‡ = ∆ ln = ∆ ln (7.89) = ∆‡ ln + 0 ª + 0 ª + 0 + 0 = ∆‡ ln (7.90) ª y + 0 ª‡ +‡ (7.91) ∆ = ∆ − ln ª + with ª = 1 bar and + = + = 1 mol l−1 . c) Thermodynamic transition state theory for reactions in the liquid phase For reactions in condensed phases, ∆ ≈ ∆ and ‡
y
¤ £ ¶ µ ¶ µ ¡ + ¢∆ ‡ ABC‡ ∆ +‡ ∆ +‡ = exp − ( ) = exp − [A] [BC] ¶ µ ¶ µ +‡ +‡ ¡ ¢∆ ‡ ∆ ∆ exp − exp − ≈ +
(7.92)
(7.93) (7.94)
• for unimolecular reactions:
µ ¶ µ ¶ ∆ +‡ ∆ +‡ ( ) = exp − exp −
(7.95)
• for bimolecular reactions:
µ ¶ µ ¶ ∆ +‡ ∆ +‡ ( ) = + exp − exp −
(7.96)
Note: The conversion between values for ∆ +‡ and ∆ ª‡ and, likewise, between values for ∆ +‡ and ∆ ª‡ for a given reaction in the liquid and gas phase requires some caution.46 46
See, e.g., D. M. Golden, J. Chem. Ed. 48, 235 (1971).
7.3 Thermodynamic interpretation of transition state theory
164
7.3.2 Applications of thermodynamic transition state theory
a) Comparison of ∆S ‡ -values for different types of transition states I
Bimolecular reactions: y
A + BC → ABC‡
(7.97)
∆ ‡ 0
(7.98)
∆ ‡ ¿ 0
(7.99)
or even Furthermore, for chemically otherwise similar molecules (nonlinear molecule) (linear molecule) À (cyclic molecule)
(7.100)
Thus, ¯ ‡ ¯ ¯ ¯ ¯ ¯ ¯∆ (nonlinear molecule)¯ ¯∆ ‡ (linear molecule)¯ ¿ ¯∆ ‡ (cyclic molecule)¯ (7.101) For example: ¯ ¯ ⎛ ⎞¯ ¯ ⎛ ⎛ ‡ ¯ ¯ ¯ ‡ ⎞¯¯ ¯ A · · · B A ··· ¯ ¯ ¯ ¯ ¯ ¯ ¯ ‡⎜ ¯ ⎟ ⎜ . ‡ ‡ . ¯ ¯ .. ¯∆ ⎝ ⎠¯ ¯∆ ⎝ A · · · B · · · C ⎠¯ ¿ ¯∆ ⎝ .. ¯ ¯ ¯ ¯ ¯ ¯ ¯ C ¯ C ··· y
I
⎞¯ ‡ B ¯¯ .. ⎟¯ . ⎠¯¯ D ¯ (7.102)
cyclic ¿ linear nonlinear
(7.103)
ABC → ABC‡
(7.104)
Unimolecular reactions:
• For isomerization reactions with a tight transition state: ⎛ ‡⎞ B ⎠.0 Â ∆ ‡ ⎝ Á A – C
(7.105)
• For bond fission reactions with a loose transition state: ¡ ¢ ∆ ‡ AB · · · C‡ 0
(7.106)
y loose tight
(7.107)
7.3 Thermodynamic interpretation of transition state theory
165
b) Thermochemical estimation of A-factors* Equations 7.71 and 7.86 allow us to estimate the -factors of chemical reaction rate constants by the following procedure (Benson1976): (1) We setup a model for the TS, with a certain structure and vibrational freuqencies. (2) We take a reference molecule that is similar to the TS and use the ref of the reference molecule for a zero-order estimation of ∆ ‡ according to X ( reactants ) (7.108) ∆ ‡ = ref − reactants
(3) We apply corrections for the differences between ref and ‡ based on the statistical thermodynamics result of = ln +
ln
(7.109)
y
µ ‡¶ ‡ (7.110) ln ∆ = − = ln ref + ref Usually, the correction term accounts for differences of the entropies for translation, rotation, internal rotations, vibrations, symmetry, electron spin, plus sometimes other terms (e.g., optical isomers). cor
I
I
‡
ref
Example 7.1: Estimation of the -factor for the reaction O + CH4 (O · · · H · · · CH3 )‡ → OH + CH3 .
→
∆ ‡ = (O · · · H · · · CH3 )‡ − (O) − (CH4 )
(7.111)
ref = (CH3 F)
(7.112)
∆ cor thus depends on a comparison of CH3 F and OHCH‡3 (see Table ).
¤
Table 7.4: Estimation of the -factor for the reaction O + CH4 (O · · · H · · · CH3 )‡ → OH + CH3 .
→
∆ ‡ ( J mol−1 K−1 ) −1243 + 91
Contribution ª (CH3 F) − ª (O) − ª (CH4 ) Spin: + ln 3 1 (1 2 3 )‡ external rotation: ln 3 (1 2 3 )ref translation CF = 1000 cm−1 → OH = 2000 cm−1 −1 2 OHC () ³ = 600 cm ´
+
79
− − + +
04 04 00 42
∆ ‡ = ª O · · · H · · · CH‡3 − ª (O) − ª (CH4 ) −1039
Correction ∆‡ → ∆‡ : + ln (0 )
y
∆ ‡ = −198 J mol−1 K−1 .
7.3 Thermodynamic interpretation of transition state theory
166
Estimated -value: =
µ
∆ ‡ exp
2
¶
(7.113) µ
−197 J mol−1 K−1 = 739 × 625 × 10 × exp 831 J mol−1 K−1 = 43 × 1012 cm3 mol−1 s−1 12
¶
cm3 mol−1 s−1 (7.114) (7.115)
Experimental -value: expt = 19 × 1012 cm3 mol−1 s−1
(7.116)
The difference is probably due to a poor estimate of the TS structure — one can do much better, I just didn’t try hard enough here.47
7.3.3 Kinetic isotope effects
I
Figure 7.5: The kinetic isotope effect due to the difference in the zero-point energies of the reactants and the respective TS structures.
47
For H abstraction reactions from a series of hydrocarbons by 3 CH2 , the agreement that was achieved was better than ±30 %.
7.3 Thermodynamic interpretation of transition state theory
167
Kinetic isotope effects occur in particular in the case of H/D atom transfer reactions. The reason is that the threshold energy ∆0‡ depends on the vibrational zero-point energies. We consider the reactions A + H—R → A—H—R‡ → products A + D—R → A—D—R‡ → products
(7.117) (7.118)
∆0‡ (H—R) = ∆ ‡ + ‡ (A—H—R) − (H—R)
(7.119)
where ∆0‡ (D—R)
= ∆ ‡ + ‡ (A—D—R) − (D—R)
with = y
1X 2
(7.121)
‡ = ∆0‡ (D—R) − ∆0‡ (H—R) ∆0(DH)
(7.122)
= +‡ (A—D—R) − ‡ (A—H—R) − ( (D—R) − (H—R))
y
I
(7.120)
à ! ‡ ∆0(D/H) (D—R) = exp − (H—R)
(7.123)
(7.124)
Example 7.2: R-H/R-D substitution. The vibrational frequencies depend on the force constants and reduced masses : 1 = 2
µ ¶12
(7.125)
For H/D substitution: 1 (HR) ≈ (DR) 2 y
(HR) = (DR)
µ
(HR) (DR)
¶12
(7.126)
≈
√ 2
(7.127)
With (HR) = 3000 cm−1 and (DR) = 2100 cm−1 , we find ∆ = 1 (3000 − 2100) cm−1 = 450 cm−1 , by which the DR reactant is stabilized compared 2 to the HR reactant. For = 300 K: µ ¶ (DR) 450 (7.128) = exp − ≈ −225 ≈ 01 (HR) 200 ¤ For more realistic estimates, one has to take into account the change of all vibrational frequencies in the reacting molecules and the TS.
7.3 Thermodynamic interpretation of transition state theory
168
7.3.4 Gibbs free enthalpy correlations I
Figure 7.6: Linear free enthalpy correlations.
Application of the TST expression
µ ¶ ∆‡ exp − =
(7.129)
to a series of related reactions (1) and (2) often gives a relation of the type48 ¡ ¢ ª ∆‡1 − ∆‡2 = ∆ª (7.130) 1 − ∆2
y
ln
1 1 = ln 2 2
(7.131)
7.3.5 Pressure dependence of k Liquid phase reactions often show a pressure dependence of the rate constant. Using the relation ¶ µ = (7.132) we can predict this pressure dependence. Starting with µ ¶ ∆‡ = exp − (7.133) we find
48
µ
ln
¶
=−
∆ ‡
A well-known example is the Hammett correlation in organic chemistry.
(7.134)
7.3 Thermodynamic interpretation of transition state theory
169
I
Activation volume:
I
Example 7.3: Comparison of the pressure dependencies of the rate constants for 1 and 2 reactions:
∆ ‡
(7.135)
(CH3 )3 CCl + H2 O → (CH3 )3 C+ + Cl− + H2 O →
HO− + CH3 I
⎤− | → ⎣ HO · · · C · · · I ⎦ ÁÂ ⎡
∆ ‡ 0
y
↑ y
→
∆ ‡ 0
y
↑ y
↓
1
↑
2 ¤
7.4 Transition state spectroscopy
170
7.4 Transition state spectroscopy Until a few years ago, the “transition state” was a rather “elusive and strange” species. This situation has completely changed (a) because of the rapidly improving quantum chemical methods for computing TS properties and (b) because of transition state spectroscopy, in the frequency domain (J. Polanyi, Kinsey), and in the time domain (Zewail). I
Figure 7.7: Transition state spectroscopy: The reaction K + NaCl.
I
Figure 7.8: Transition state spectroscopy: Raman emission of dissociating molecules.
7.4 Transition state spectroscopy
I
Figure 7.9: Transition state spectroscopy: The reaction F + H2 .
I
Figure 7.10: Transition state spectroscopy: NaI Photodissociation.
171
7.4 Transition state spectroscopy
172
I
Figure 7.11: Transition state spectroscopy: NaI Photodissociation.
I
Figure 7.12: Transition state spectroscopy: Reaction a in van-der-Waals Complex.
7.4 Transition state spectroscopy
I
Figure 7.13: Transition state spectroscopy: Stimulated Emission Pumping.
173
7.4 Transition state spectroscopy
174
7.5 References Benson 1976 S. W. Benson, Thermochemical Kinetics, Methods for the Estimation of Thermochemical Data and Rate Parameters, 2nd Ed., Wiley, New York, 1976. Evans 1935 M. G. Evans, M. Polanyi, Trans. Faraday Soc. 31, 875 (1935). Eyring 1935 H. Eyring, J. Chem. Phys. 3, 107 (1935). Herschbach 1987 D. R. Herschbach, Molekulardynamik einfacher chemischer Reaktionen, Angew. Chem. 99, 1251 (1987). Lee 1987 Y. T. Lee, Molekularstrahluntersuchungen chemischer Elementarprozesse, Angew. Chem. 99, 967 (1987). Polanyi 1987 J. C. Polanyi, Konzepte der Reaktionsdynamik, Angew. Chem. 99, 981 (1987). Wigner 1938 E. Wigner, Trans. Faraday Soc. 34, 29 (1938).
7.5 References
175
8. Unimolecular reactions Unimolecular reactions are reactions in which only one species experiences chemical change. I
Figure 8.1: Examples of unimolecular reactions.
8.1 Experimental observations (1) There are many reactions governed by a first-order rate law, [A] = −∞ [A] ,
(8.1)
for example isomerization and dissociation reactions like Cyclopropane -Butene C2 H6 Cycloheptatriene
→ → → →
Propene -Butene 2 CH3 Toluene
(8.2) (8.3) (8.4) (8.5)
(2) In these reactions, no other species than the reactants experience any chemical change. (3) These reactions have the same rates in the gas phase and in solution, gas
phase
≈ solution
phase
,
i.e., the environment has no effect on the reaction rates!
(8.6)
8.1 Experimental observations
176
(4) There are many similar other reactions, however, for which the rate law is secondorder, i.e., [A] (8.7) = −0 [A] [M] Examples are Br2 → Br + Br H2 O2 → OH + OH
(8.8) (8.9)
(5) More detailed measurements show that at a given temperature the order of the reaction may depend on pressure (see Fig. 8.2), for instance for the reaction CH3 NC → CH3 CN
(8.10)
one has found that a) for → 0:
[CH3 NC] = −0 [CH3 NC] [M] This is called the “low pressure range” for the reaction.
b) for → ∞:
[CH3 NC] = −∞ [CH3 NC] This is called the “high pressure range” for the reaction”.
(8.11)
(8.12)
(6) The transition from the low to the high pressure range depends a) on the size of the molecule (see Fig. 8.3), b) for a given molecule, it depends on temperature. (7) The activation energy in the low pressure regime is much smaller than in the high pressure regime: ln (8.13) = 2 where 0 ∞ (8.14) and ∞ ≈ 0
(8.15)
(8) The preexponential factors in the low pressure regime are much higher than the collision frequencies, i.e., (8.16) 0 (9) For all these reactions, the molecules must somehow be “energized”. There are other reactions, in which the molecules are energized chemically (“chemically activated reactions”) or by photons. At the microscopic, i.e., molecular level, these reactions should have related mechanisms.
8.1 Experimental observations
I
Questions: (1) What are the differences of the reactions under (1), (4) and (5)? (2) How can we rationalize their reaction mechanisms?
I
Figure 8.2: Schematic fall-off curve of the unimolecular rate constant.
I
Figure 8.3: Experimentally observed fall-off curves of unimolecular reactions.
177
8.2 Lindemann mechanism
178
8.2 Lindemann mechanism First attempts to explain the pressure dependence of unimolecular reaction rate constants due to radiation and the dissociation dynamics due to centrifugal forces (rotational excitation) proved to be wrong. In addition, the historic examples for unimolecular reactions, for instance the N2 O5 decomposition, were later proven to have more complex reaction mechanisms. The first “realistic” model that explained the pressure dependence of was that proposed by Lindemann (1922). I
Figure 8.4: Lindemann mechanism.
I
Figure 8.5: Lindemann mechanism: High pressure regime.
8.2 Lindemann mechanism
I
Figure 8.6: Lindemann mechanism: Low pressure regime.
I
Figure 8.7: Fall-off curve: The proper way to plot log is vs. log [M].ab
a
b
179
Since 0 ∝ [M], we can also plot log vs. log 0 . This gives the so-called doubly reduced fall-off curves. In reality, the fall-off regime extends over several orders of magnitude in [M].
8.2 Lindemann mechanism
I
180
Half-pressure: We have found the following results: 2 1 [M] −1 [M] + 2
(8.17)
1 −1
(8.18)
0 = 1 [M]
(8.19)
=
∞ = 2 Thus, if −1 [M] = 2 , we have =
1 2 1 2 1 [M] 2 1 [M] 1 = = = ∞ −1 [M] + 2 2−1 [M] 2 −1 2
(8.20)
Half “pressure”: [M]12 =
I
2 −1
(8.21)
Lifetimes of energized molecules: It is instructive to look at the reciprocal values: 1 1 = −1 [M] 2
(8.22)
•
1 is the time between two collisions. This is, in fact, the lifetime of the −1 [M] energized molecules with respect to collisional deactivation.
•
1 is the lifetime of the excited molecules with respect to reaction. 2
Thus, at the half pressure, we have Collisional Deactivation = Unimolecular Decay
I
Determination of k∞ : = y 1
2 1 [M] −1 [M] + 2
−1 1 1 + 1 2 1 [M] 1 1 1 + = ∞ 1 [M] 1 1 + = ∞ 0 =
(8.23)
(8.24)
(8.25) (8.26) (8.27)
y Plot of −1 vs. [M]−1 should give a straight line with slope 1 −1 and intercept ∞ −1 .
8.2 Lindemann mechanism
181
I
Deficiencies of the Lindemann model:
I
Figure 8.8:
• Activation energies:
0 ∞
(8.28)
∞ ≈ 0
(8.29)
0 ≈ 0 − ( − 15)
(8.30)
In particular:
8.3 Generalized Lindemann-Hinshelwood mechanism
8.3 Generalized Lindemann-Hinshelwood mechanism 8.3.1 Master equation
I
Figure 8.9: Strong collision model.
I
Figure 8.10: Generalized Lindemann-Hinshelwood weak collision model.
182
8.3 Generalized Lindemann-Hinshelwood mechanism
183
I
Figure 8.11: Generalized Lindemann-Hinshelwood weak collision model.
I
Figure 8.12: Generalized Lindemann-Hinshelwood model: Rate constants in the high and low pressure (strong and weak collision) limits.
8.3 Generalized Lindemann-Hinshelwood mechanism
184
8.3.2 Equilibrium state populations As seen from the master equation analysis, the unimolecular reaction rate constant in the high and in the low pressure regimes depends on the equilibrium (Boltzmann) state distributions (for discrete states) or () (continuous state distributions). In a polyatomic molecule, due to the large number of vibrational degrees of freedom, the number of vibrational states increases very rapidly with increasing vibrational excitation energy. The quanta of vibrational excitation can be distributed over the oscillators. In order to calculate the number of combinations, we start with a single oscillator and expand our scope first to two oscillators and then to = 3 − 6 oscillators. a) Equilibrium state population for a single oscillator: I
Boltzmann distribution: =
−
(8.31)
with the energy quanta = × ;
the degeneracy factor
= 0 1 2
(8.33)
and the partition function = I
(8.32)
1 1 − −
(8.34)
Limiting case for high T (transition to classical mechanics): ¿ y = classical
(8.35)
since − ≈ 1 − for → 0
(8.36)
Note that this classical description of vibrations is not so bad for unimolecular reactions, where we are interested in the kinetic behavior at high temperatures. b) Equilibrium state population for s oscillators:
y
() → () () =
() −
(8.37)
(8.38)
with () being the density of vibrational states at the energy and =
Y
1
1− =1
−
(8.39)
and = 3 − 6
(resp. = 3 − 5)
(8.40)
8.3 Generalized Lindemann-Hinshelwood mechanism
185
8.3.3 The density of vibrational states ρ (E) The density of vibrational states is defined as () =
number of states in energy interval energy interval
(8.41)
()
(8.42)
i.e., () =
In a polyatomic molecule, as we shall see in the following, () increases with very rapidly owing to the large number of overtone and combination states, and reaches truly astronomical values: a) s = 1: () =
1
(8.43)
b) s = 2: We consider the ways, in which we can distribute two identical energy quanta between the two oscillators: Osc. 1 Osc. 2 Osc. 1 Osc. 2 Osc. 1 Osc. 2 Osc. 1 Osc. 2 0 1 | | 2 2 || | | || 3 3 ||| || | | || ||| 4 4 5 5 6 .. .. .. .. . . . .
(8.44)
c) s oscillators: • For simplicity, we consider a molecule with identical harmonic oscillators. • This molecule shall be excited with quanta, each of size , i.e., the molecule has the excitation energy = I
Question: How can we distribute these quanta over the oscillators?
8.3 Generalized Lindemann-Hinshelwood mechanism
I
186
Answer: We are asking for the number of distinguishable possibilities to distribute quanta between oscillators, (). This is identical to the problem of distributing balls between boxes, for example for = 11 and = 8: ••• • •
• •• • ••
(8.45)
That number is identical to the number of distinguishable permutations of balls ( • ) and − 1 walls ( | ), ( + − 1)! () = (8.46) ! ( − 1)! where ( + − 1)! is the total number of permutations, which is corrected by dividing through ! ( − 1)! to account for the number of indistinguishable permutations among the balls (!) and walls (( − 1)!) among each other (which give indistinguishable results). I
Answer: We make the following approximation for À :49 ( + − 1)! −1 ≈ ! ( − 1)! ( − 1)!
I
Question: How can we compute ()?
I
Answer:
(8.49)
()
(8.50)
∆ =
(8.51)
= ×
(8.52)
() = (1) We consider the energy interval
at the excitation energy In this energy interval, we have the number of states ∆() = () y () =
49
∆ () 1 ( + − 1)! () ≈ = ∆ ! ( − 1)!
(8.53)
(8.54)
The approximation ( + − 1)! ≈ −1 !
(8.47)
can be derived using Stirling’s formula (ln ! = ln − ) and the power series expansion of ln (1 − ) ≈ for || ¿ 1 (see homework assignment).
(8.48)
8.3 Generalized Lindemann-Hinshelwood mechanism
(2) Using the approximation
187
−1 ( + − 1)! ≈ (Eq. 8.49) for À gives ! ( − 1)! ( − 1)! () =
1 −1 ( − 1)!
(8.55)
(3) Inserting = into (), we obtain 1 ()−1 () = ( − 1)!
(8.56)
−1 ( − 1)! ()
(8.57)
(4) Result: () =
d) Notes:* (1) The above derivation may seem artificial because in a real molecule not all oscillators are identical. However, the derivation can be easily generalized. (2) For instance, we can start from the number of states of the first oscillator 1 (1 ) and convolute this with the density of states 2 () of the second oscillator 2 (2 ) = 2 ( − 1 ) =
1
(8.58)
and compute the total (combined) density of states for two oscillators by convolution according to Z () = 1 (1 ) 2 ( − 1 ) 1 (8.59) 0
etc. up to oscillators. The result is () =
−1 Q ( − 1)! =1
(8.60)
(3) We can also use inverse Laplace transforms: Since is the Laplace transform of () according to Z ∞ = () − = L [ ()] (8.61) 0
we can determine () from (which we know) by inverse Laplace transformation. (4) With corrections for zero-point energy: ( + )−1 Q () = ( − 1)! =1
(8.62)
8.3 Generalized Lindemann-Hinshelwood mechanism
188
(5) With (empirical) Whitten-Rabinovitch corrections: () =
( + () )−1 Q ( − 1)! =1
(8.63)
where () is a correction factor that is of the order of 1 except at very low energies. (6) Exact values of () for harmonic oscillators can be obtained by direct state counting algorithms. (7) Corrections for anharmonicity can be applied using different means.
8.3.4 Unimolecular reaction rate constant in the low pressure regime From the master equation analysis above, we obtained the thermal unimolecular reaction rate constant in the low pressure regime as XX −1() [M] (8.64) 0 = The reduction of the excited state populations compared to the equilibrium (Boltzmann) distribution is shown in Fig. 8.13. Two cases can be distinguished: (1) With the strong collision assumption (h∆ i À ), we can apply the so-called equilibrium theories which assume that = for ≤ 0 . Thus, the state population below 0 remains as in the high pressure regime, whereas it is reduced above 0 .Then, XX XX X −1() [M] = −1() [M] = [M] (8.65) 0 = y
Z∞ 0 = [M] ()
(8.66)
0
Using the expressions for () () and in the classical limit ( ¿ ) derived above and doing the integration by parts, we obain the Hinshelwood equation for 0 : µ ¶−1 1 0 0 = [M] (8.67) −0 ( − 1)! This equation is usually used to describe experimental data with as a fit parameter (usually about half the real ). Since ∝ 05 and = 2 ln , the low pressure activation energy becomes 0 = 0 − ( − 15) (8.68) Thus, may be significantly smaller than 0 . This can be understood as illustrated in Fig. 8.14.
8.3 Generalized Lindemann-Hinshelwood mechanism
189
(2) With the weak collision assumption (h∆ i ≤ ) made by the so-called nonequilibrium theories, the bottleneck in the state population falls below the equilibrium distribution already below 0 (see Fig. 8.14). Using the weak collision assumption, Troe derived the expression 0 = [M]
(0 ) −0
(8.69)
where is the so-called collision efficiency ( ≤ 1) which can be expressed in terms of the average energy transferred per collision h∆ i. becomes even smaller than the Hinshelwood above. A simple estimate of the weak collision effect based on Fig. 8.14 may lead to 0 ≈ 0 − ( − 05)
I
Figure 8.13: State populations.
(8.70)
8.3 Generalized Lindemann-Hinshelwood mechanism
I
190
Figure 8.14: Activation energies for unimolecular reactions in the high pressure, the low pressure strong collision, and the low pressure weak collision limits.
8.3.5 Unimolecular reaction rate constant in the high pressure regime From the master equation analysis above, the thermal unimolecular reaction rate constant in the high pressure regime is (see Fig. 8.12)
∞
Z∞ = () ()
(8.71)
0
with the Boltzmann distribution () =
() −
(8.72)
as shown in Fig. 8.13. Averaging over the Boltzmann distribution gives the TST expression ( ) =
‡ −0
(8.73)
8.4 The specific unimolecular reaction rate constants ()
191
8.4 The specific unimolecular reaction rate constants k (E) We may intuitively expect that the specific uinimolecular reaction rate constants () depend on the energy content (i.e., the excitation energy ) of the energetically activated molecules molecules A∗ . More precisely, we will also have to take into account the dependence of the specific rate constants on other conserved degrees of freedom, in particular total angular momentum (⇒ ( )), and perhaps other quantities (symmetry, components of if -mixing is weak, (⇒ ( ; ; ))).
8.4.1 Rice-Ramsperger-Kassel (RRK) theory I
RRK model: In order to derive an expression for the specific rate constants (), we realize that it is not enough that the molecules are energized to above 0 . Indeed, for the reaction to take place, the energy also has to be in the right oscillator, namely in the RC. In particular, for the reaction to take place, of the total excitation energy , the amount † has to be concentrated in the RC such that † ≥ 0 . Thus, we can write the reaction scheme as †
∗ → † →
(8.74)
where † denotes those excited molecules that have enough energy in the RC to overcome 0 . We assume that once this critical configuration († ) is reached, the molecules will immediately cross the TS and go to the product side. I
Probabilities of A† and A∗ : The probability of † compared to ∗ can be derived using statistical arguments by considering the ratio of the density of states. We consider a molecule with • oscillators, • = quanta which have to be distributed over the oscillators, • a threshold energy of 0 such that a minimum of = 0 quanta have to be concentrated in the RC and only − quanta can be distributed freely, • À and − À . The probability of † relative to ∗ is given by ¡ ¢ ¡ ¢ † † = (∗ ) (∗ ) where (∗ ) ∝ ∗ () =
1 ( + − 1)! ! ( − 1)!
(8.75)
(8.76)
and, since in the critical configuration only − quanta can be distributed freely, ¡ ¢ 1 ( − + − 1)! † ∝ † () = ( − )! ( − 1)!
(8.77)
8.4 The specific unimolecular reaction rate constants ()
y
I
¡ ¢ † ( − + − 1)! ! ( − 1)! = × ∗ ( ) ( − )! ( − 1)! ( + − 1)! ( − + − 1)! ! = × ( − )! ( + − 1)!
(8.78) (8.79)
Specific rate constant: The critical configuration † is reached with the rate coefficient (≡ frequency factor) † (see the reaction scheme above). Since the energy is now in place (in the RC), † will cross the TS to products. The specific rate constant () is therefore simply ¡ †¢ () = † × (∗ ) y () = † ×
I
192
! ( − + − 1)! × ( − )! ( + − 1)!
(8.80)
(8.81)
Kassel formula: With the approximations according to Eq. 8.49),
and
we obtain
y
( + − 1)! ≈ −1 for À !
(8.82)
( − + − 1)! ≈ ( − )−1 for − À , ( − )!
(8.83)
¡ ¢ µ ¶−1 † − ( − )−1 = = (∗ ) −1 ¡ ¢ µ ¶−1 † − 0 = (∗ )
(8.84)
(8.85)
so that the specific rate constant becomes () =
†
µ
− 0
¶−1
(8.86)
This expression is known as the Kassel formula for (). I
Notes on the Kassel formula: • Owing to the approximations À and − À , the Kassel formula is valid only at À 0 .
8.4 The specific unimolecular reaction rate constants ()
193
• The Kassel formula describes experimental data qualitatively well. However, instead of the full value of = 3 − 6, one usually has to use an effective number eff . As suggested by Troe, eff can be estimated from the specific heat of the reactant molecules via µ ¶ µ ¶ 2 ln = ≈ eff = (8.87) using the known expression for , =
Y
1 1 − exp (− ) =1
(8.88)
• Often, eff is equal to about one half of the number of oscillators, eff ≈ I
(3 − 6) 2
(8.89)
Figure 8.15: Specific rate constants: RRK formula.
8.4.2 Rice-Ramsperger-Kassel-Marcus (RRKM) theory RRK theory fails in the following points: • It can be used only with as an effective parameter (eff ). • It leads to the wrong results close to threshold, where → 0 (see the conditions above). • It is difficult to derive for oscillators with different frequencies. • RRK theory is essentially a classical theory. It does not know about zero-point energy.
8.4 The specific unimolecular reaction rate constants ()
I
194
RRKM expression: These problems are essentially solved by RRKM theory which gives the expression ‡ ( − 0 ) () = (8.90) () where • ‡ ( − 0 ) is the number of open channels (essentially the number of states at the transition state configuration exluding the RC), counted from the zero-point level of the TS, at the energy − 0 , • () is the density of states of the reacting molecules at the energy . These quantities are the two critical quantities in microcanonical transition state theory.
I
Figure 8.16: RRKM expression for ().
8.4 The specific unimolecular reaction rate constants ()
195
I
Figure 8.17: RRKM expression for ( ) with angular momentum conservation.
I
Figure 8.18: The sum and the density of states.
8.4 The specific unimolecular reaction rate constants ()
I
RRKM expression with angular momentum conservation: Eq. 8.90 can be generalized to include other conserved quantities, e.g., total angular momentum, ( ) =
I
196
‡ ( − 0 ) ( )
(8.91)
Relation between RRKM and RRK theory: To elucidate the relation between Eqs. 8.86 and 8.90, we look at () and ( − 0 ) more closely. (1) Density of vibrational states (): () =
number of states in energy interval () = energy interval
(8.92)
a) RRK result for the number of indistinguishable permutations of quanta À : over oscillators; = 1 −1 1 ()−1 1 ( + − 1)! ≈ = () = ! ( − 1)! ( − 1)! ( − 1)! y () =
−1 ( − 1)! ()
(8.93)
(8.94)
b) It is relatively easy to show that for oscillators with different frequencies, the expression for () becomes () =
−1 Q ( − 1)! =1
(8.95)
c) With corrections for zero-point energy: () =
( + )−1 Q ( − 1)! =1
(8.96)
d) With (empirical) Whitten-Rabinovitch correction:
( + () )−1 Q () = ( − 1)! =1
(8.97)
where () is a correction factor that is of the order of 1 except at very low energies. e) Exact values of () for harmonic oscillators can be obtained by direct state counting algorithms. Corrections for anharmonicity can be applied using different means.
8.4 The specific unimolecular reaction rate constants ()
197
(2) Number of states () and ‡ ( − 0 ): Since () = () , the number of states from = 0 to can be obtained by integration, () =
Z
(8.98)
()
0
a) RRK expression:
() =
Z
() =
0
Z 0
−1 1 1 = ( − 1)! () ( − 1)! ()
Z
−1
0
(8.99) y () =
! ()
(8.100)
b) Allowing for different frequencies, we modify this expression to () =
!
Q
=1
(8.101)
c) As above, a correction for zero-point energy gives () =
( + ) Q ! =1
(8.102)
d) With the Whitten-Rabinovitch correction, this becomes () =
( + () ) Q ! =1
(8.103)
e) Correction for → 0: At = 0, we must have at least one state, i.e., ‡ () = 1 +
( + () ) ( (0) ) Q − Q ! =1 ! =1
(8.104)
(3) Application to unimolecular reactions: a) At the TS, we have only − 1 oscillators since the RC has to be excluded. Furthermore, the states range from the zero-point energy 0 of the TS to , while we count from the zero-point of the reactants. y ³ ³ ´−1 ´−1 − 0 + ‡ ( − 0 ) ‡ ‡ (0) ‡ ‡ ( − 0 ) = 1 + − Q Q ‡ ‡ ( − 1)! −1 ( − 1)! −1 =1 =1 (8.105) ‡ This expression gives the correct value of ( − 0 ) = 1 at = 0 .
8.4 The specific unimolecular reaction rate constants ()
198
b) Density of states: () = c) Specific rate constant:
( + () )−1 Q ( − 1)! =1
(8.106)
³ ³ ´−1 ´−1 ‡ ‡ ‡ ‡ − 0 + ( − 0 ) (0) 1 + () = − Q−1 ‡ Q ‡ () () ( − 1)! =1 () ( − 1)! −1 =1 (8.107) (4) For energies sufficiently above 0 , we can use the expressions ³ ´−1 − 0 + ‡ ‡ ( − 0 ) = Q ‡ ( − 1)! −1 =1
and
( + )−1 Q () = ( − 1)! =1
to obtain
³ ´−1 ‡ Q − + 0 ( − 0 ) =1 = Q−1 () = ‡ () ( + )−1 =1 ‡
y
à !−1 Q ‡ + − 0 =1 () = Q−1 ‡ + =1
(8.108)
(8.109)
(8.110)
(8.111)
The resemblence to the Kassel formula is obvious.
8.4.3 The SACM and VRRKM model More advanced models allow for the tightening of the TS with increasing (variational RRKM theory = VRRKM theory) or even for individual adiabatic channel potential curves (statistical adiabatic channel model = SACM). The latter model has to be used for reactions with loose transition states.
8.4 The specific unimolecular reaction rate constants ()
199
8.4.4 Experimental results
I
Figure 8.19: Experimental methods for the preparation of highly vibrationally excited molecules with defined excitation energy.
I
Figure 8.20: Comparison of theoretically predicted specific rate constants () for unimolecular isomerization of cycloheptatrienes with directly measured experimental data.
8.5 Collisional energy transfer*
8.5 Collisional energy transfer* Skipped for lack of time.
200
8.6 Recombination reactions
8.6 Recombination reactions See homework assignment.
201
8.6 Recombination reactions
202
8.7 References Lindemann 1922 F. A. Lindemann, Trans. Faraday Soc. 17, 598 (1922). Hinshelwood 1927 C. N. Hinshelwood, Proc. Roy. Soc. A113, 230 (1927). Kassel 1928a J. Phys. Chem. 32, 225 (1928). Kassel1928b J. Phys. Chem. 32, 1065 (1928). Kassel 1932 L. S. Kassel, The Kinetics of Homogeneous Gas Reactions, Chem. Catalog, New York, 1932. Rice 1927 O. K. Rice, H. C. Ramsperger, J. Am. Chem. Soc. 49, 1617 (1927). Rice 1928 O. K. Rice, H. C. Ramsperger, J. Am. Chem. Soc. 50, 617 (1928). Troe1979 J. Troe, J. Phys. Chem. 83, 114 (1979). Troe1994 J. Chem. Soc. Faraday Trans. 90, 2303 (1994).
8.7 References
203
9. Explosions 9.1 Chain reactions and chain explosions 9.1.1 Chain reactions without branching I
Reaction scheme: (1) A →X chain initiation (2) X + B → P + X chain propagation (4) X →A chain termination
I
Rate laws: [X] = 1 [A] − 4 [X] [P] = 2 [B] [X]
(9.1) (9.2)
As we see, reaction (2) does not appear in the rate law because → ! I
Solutions of Eq. 9.1 for two limiting cases (cf. Fig. 9.1): (1) Solution for → 0 and [A] ≈ const by substitution, using = 1 [A] − 4 [X] y
= −5 y [X] = − [X] 4
y − y
1 =y = −4 4
= −4 = 1 [A] − 4 [X]
y [X] =
1 [A] − −4 4
(9.4)
1 [] 4
(9.5)
= 0: [X] = 0 y = y [X] =
(9.3)
´ 1 [A] ³ 1 − −4 4
(9.6)
9.1 Chain reactions and chain explosions
204
(2) Solution with steady state assumption for [X] at À (i.e, after the induction time ): [X] ≈0 (9.7) y 1 [X] = [A] (9.8) 4 y Considering the free radical concentration profile [X ()], no desasters are happening.
I
Figure 9.1: Evolution of the free radical concentration in a normal chain reaction.
9.1.2 Chain reactions with branching I
Reaction scheme: (1) A →X chain initiation (3) X + B → P + X chain branching = branching factor (4) X →A chain termination
I
Most important example for a chain branching reaction: H + O2 → OH + O
(9.9)
This reaction is responsible for flame propagation in all hydrocarbon flames: Since H is small, it has the highest diffusion coefficient and diffuses rapidly into the unburnt gases.
9.1 Chain reactions and chain explosions
I
205
Rate laws: [X] = 1 [A] + ( − 1) 3 [X] [B] − 4 [X] [P] = 3 [B] [X]
I
(9.10) (9.11)
Solutions for three limiting cases: (1) = 1 or 4 3 ( − 1) [B]:
[X] ≈0
y [X] = y
1 [A] 4 − ( − 1)3 [B]
(9.12) (9.13)
[P] (9.14) = 3 [B] [X] = The overall reaction is stable, because the radical concentration is stable. The product formation rate is final (constant, well behaved). (2) 4 = 3 ( − 1) [B]: At short times, [A] [B] ≈ const y [X] = 1 [A] = const
(9.15)
[X] → ∞
(9.16)
y The overall reaction turns unstable, because the radical concentration goes to infinity and thus the product formation rate too, leading to chain explosion! (3) 4 3 ( − 1) [B]: At short times, [A] [B] ≈ const y [X] + 4 [X] − 3 ( − 1) [X] [B] = 1 [A] [X] + [X] (4 − 3 ( − 1) [B]) = 1 [A] y
³ ´ 1 [A] × exp [( − 1) 3 [B] ] − 1 [X] = ( − 1) 3 [B] − 4
(9.17) (9.18)
(9.19)
The overall reaction turns unstable, because the radical concentration increases exponentially, leading to an exponentially growing product formation rate, and therefore chain explosion!
9.1 Chain reactions and chain explosions
I
206
Figure 9.2: Evolution of the free radical concentrations in a chain reaction with branching (three limiting cases).
9.2 Heat explosions
207
9.2 Heat explosions We consider an exothermic bimolecular reaction in a closed reactor ∆ ª 0
A+B→C+D
(9.20)
and look at the radial temperature profile in the reactor. I
Figure 9.3: Temperature profiles in a stirred and in an unstirred reactor.
9.2.1 Semenov theory for “stirred reactors” (T = const) I
Heat production in an exothermic reaction (∆ H ª < 0): • Energy released per volume:
£
∆ ∆ ª • Reaction rate:
kJ cm3
¤
= − [] []
• heat release:
ª = ∆ 0 − | {z }
(9.21)
(9.22)
= ∆ ª
(9.23) | {z }
Arrhenius rate constant in pressure units
(9.24)
9.2 Heat explosions
208
• positive feedback effect:
reaction rate increases y temperature ↑ y reaction rate ↑ y heat explosion !!
I
Figure 9.4: Heat explosions.
I
Heat removal: At constant temperature in the stirred reactor, we use Newton’s law
= ( − 0 )
where 0
: : : :
heat conduction coefficient (depends on material) reactor surface area temperature wall temperature
(9.25) (9.26)
As we see, the heat removal increases (becomes steeper), when we increas the reactor surface area (at constant reactor volume). I
Stability limits: (1) 0 = 1 : Heat removal wins over heat production y negative feedback y stable. (2) 0 = 2 : Heat removal wins over heat production y negative feedback y stable. (3) 0 = 3 : Heat production wins over heat removal y positive feedback y heat explosion, but the system should never reach point [3]. (4) 0 = 4 : tangent y stability limit, the system runs away upon the smallest perturbation! (5) 0 = 5 : always unstable!
9.2 Heat explosions
I
209
Conclusions: • The quantity that determines whether the system is stable or not is the wall temperature 0 , because 0 determines the initial point of the heat removal line. • Smaller surface-to-volume ratio is dangerous, because smaller surface at a given volume leads to a smaller slope of the heat removal line y instability! • Security measure: internal cooling system (cold water pipes, heat exchanger). • Emergency measure: when cooling system fails, the reaction mixture should be strongly diluted by addition of solvent immediately!
I
Quantitative estimation of the stability limits:* At point (4) in Fig. 9.4, we have:
= y = y Eq y Eq y y Solution is:
( − 0 ) = ∆ ª 0 − = ∆ ª 0 − 2 2 = ( − 0 ) 2 − + 0 = 0 s µ ¶2 ± 0 − = 2 2
⎧ ´⎫ ³ p 1 ⎪ ⎪ ⎨ = + (2 − 40 ) ⎬ 2 ³ ´ p 1 ⎪ 2 ⎭ ⎩ = − ( − 40 ) ⎪ 2
(9.27) (9.28) (9.29) (9.30) (9.31) (9.32) (9.33)
(9.34)
This can be simplified:
(1) For practically important reactions, we have: À 0 y 0 ¿
(9.35)
(2) We also can exclude unreasonably hight temperatures y only the −-sign before √ counts: y sµ ¶ 2 ∗ = − (9.36) − 0 2 2 r 40 = − 1− (9.37) 2 2 ! Ã r 40 (9.38) 1− 1− = 2
9.2 Heat explosions
210
(3) We can expand the √ : = y y y y
40 ; 1; √ 1 2 1 − 1−=1− − 2·4 µ 2 ¶ 20 16 · 2 02 ∗ 1−1+ + + = 2 2 · 4 · ( )2 02 ∗ = 0 + 02 ∗ − 0 =
(9.39) (9.40) (9.41) (9.42) (9.43)
(4) Insert into in (I) and power expansion: · ·
µ
02
¶
⎡
⎤
³ ´ ⎦ · · (9.44) 0 0 1 + ¸ ∙ · 2 · · (9.45) = · ∆ ª · 0 · · exp − 0 = · ∆ ª · 0 · exp ⎣−
(5) Explosion limit: ¸ ∙ · · 2 = · exp − = ª 2 0 0 · ∆ · · · 0 · · y
∝ ·
(9.46)
(9.47)
Power expansion of the exponent in step (4): ⎤ ⎡ ¸ ∙ 0 1 1 ⎦ ⎣ ³ ´ = exp − ¿ 1 (9.48) exp − = 1+ 0 0 1 + ¸ ∙ ¢ 1¡ 2 (9.49) = exp − 1 − + ¸ ∙ 1 (9.50) ≈ exp − (1 − ) ∙ ¸ 1 = exp 1 − (9.51) ∙ ¸ 1 1 = · exp − (9.52) ¸ ∙ (9.53) = · exp − 0
9.2 Heat explosions
I
211
Figure 9.5:
9.2.2 Frank—Kamenetskii theory for “unstirred reactors”
I
Figure 9.6: Temperature profiles in an unstirred reactor.
Boundary effects are neglected in the Figure. In reality, there is a smooth transition in the boundary layer.
9.2 Heat explosions
212
9.2.3 Explosion limits
I
Figure 9.7: Explosion limits in the H2 /O2 system.
9.2.4 Growth curves and catastrophy theory I
Exponential growth: ∝ · y
I
(9.54)
Hyperbolic growth: ∝
I
=
1 y = 1+ ( − )
mit 1 + 1
(9.55)
Heat explosion:
∝
quadratic, i.e. we get hyperbolic growth !
(9.56)
9.2 Heat explosions
I
Figure 9.8: Growth curves.
213
9.2 Heat explosions
214
10. Catalysis The net rate of a chemical reaction may be affected by the presence of a third species even though that species is neither consumed nor produced by the reaction. If the rate increases, we speak of catalysis. If the rate decreases, we speak of inhibition. In general, we have to distinguish • homogeneous catalysis, • heterogeoues catalysis. 10.1 Kinetics of enzyme catalyzed reactions (homogeneous catalysis) The mechanism describing the homogeneous catalytical activity of enzymes has first been elucidated by Michaelis and Menten (1914). I
Experimental observations on enzyme catalysis: (1) For constant substrate concentration [S]0 , the reaction rate is proportional to the concentration of the enzyme [E]0 ∝ [E]0
for [S]0 = const
(10.1)
(2) At an applied enzyme concentration [E]0 , the reaction rate first increases with increasing substrate concentrations [S]0 and then reaches saturation, as is shown in Fig. 10.1. I
Figure 10.1: Enzyme catalysis: Michelis-Menten kinetics.
10.1 Kinetics of enzyme catalyzed reactions (homogeneous catalysis)
215
10.1.1 The Michaelis-Menten mechanism Basic reaction scheme:
1
E + S À ES
(10.2)
−1
ES →2 E + P
(10.3)
Typical values of 2 : 2 ≈ 102 103 s−1 Reaction rate:
(in some cases up to 106 s−1 )
[S] [P] =− = 2 [ES] [ES] = 1 [E] [S] − −1 [ES] − 2 [ES] ≈ 0
(10.4)
(10.5)
(10.6) (10.7)
Mass balance: [E]0 = y [S]0 = y
[E] + [ES] [E] = [E]0 − [ES] [S] + [ES] + [P] [S] = [S]0 − [ES] − [P]
(10.8) (10.9) (10.10) (10.11)
Steady state ES concentration: [ES] = 1 ([E]0 − [ES]) ([S]0 − [ES] − [P]) − (−1 + 2 ) [ES] ≈ 0
(10.12) (10.13)
Approximations: [E]0 ¿ [S]0 [ES] ¿ [S]0 [P] ≈ 0 for early times
(10.14) (10.15) (10.16)
y [ES] = 1 [E]0 [S]0 − 1 [S]0 [ES] − (−1 + 2 ) [ES] ≈ 0
(10.17) (10.18)
Steady state ES concentration: [ES] =
1 [E]0 [S]0 1 [S]0 + −1 + 2
(10.19)
10.1 Kinetics of enzyme catalyzed reactions (homogeneous catalysis)
216
Resulting rate expression: [P] = 2 [ES] 1 2 [E]0 [S]0 = 1 [S]0 + −1 + 2 2 [E]0 = −1 + 2 1+ 1 [S]0
=
y =
2 [E]0 1 + [S]0
(10.20) (10.21) (10.22)
(10.23)
with the Michaelis-Menten constant =
−1 + 2 1
(10.24)
Limiting cases: (1) [S]0 → ∞ y
¿ 1: y [S]0 [P] = 2 [E]0
(10.25)
[P] 2 [E]0 [S]0 =
(10.26)
=
(2) [S]0 → 0 y
À 1: y [S]0 =
Determination of 2 and from a Lineweaver-Burk plot (Fig. 10.2): 1 1 1 = + 2 [E]0 2 [E]0 [S]0 ⇒ A plot of −1 vs. [S]0 slope 2 [E]0 . y
−1
(10.27)
should give a straight line with intercept (2 [E]0 )−1 and −1 + 2 slope = = intercept 1
(10.28)
10.1 Kinetics of enzyme catalyzed reactions (homogeneous catalysis)
I
217
Figure 10.2: Lineweaver-Burk plot.
10.1.2 Inhibition of enzymes: Extended reaction mechanism: 1
E + S À ES
(10.29)
−1
ES →2 E + P
(10.30)
E + I À EI
(10.31)
3
−3
The EI complex is catalytically inactive, since I blocks the active site of E. Inhibition constant: =
3 [EI] = [E] [I] −3
(10.32)
Reaction rate (Fig. 10.2): y
[ES] = ≈ 0
(10.33)
1 1 1 = + (1 + [I]) × 2 [E]0 2 [E]0 [S]0
(10.34)
y The slope of the plot of −1 vs. [S]0 I
−1
goes up (see Fig. 10.2).
Exercise 10.1: Derive Eq. 10.34 with a reasonable assumption for the concentrations of I. ¤
10.2 Kinetics of heterogeneous reactions (surface reactions)
218
10.2 Kinetics of heterogeneous reactions (surface reactions) I
Examples for heterogeneous reactions (gas-solid interface): Many reactions, which are very slow in the gas phase, take place as heterogeneous reactions in the presence of a heterogeneous catalyst: • catalytic hydrogenation • NH3 synthesis (Haber-Bosch) • catalytic exhaust gas treatment (internal combustion engines) • catalytic DeNO processes • CO oxidation • oxidative coupling reactions of CH4 (oxide catalysts) • atmospheric reactions (at gas-solid and gas-liquid interface)
10.2.1 Reaction steps in heterogeneous catalysis Heterogeneous reactions take place via a series of steps: • Diffusion in gas (or liquid) phase to catalyst surface. • Adsorption: — “physical” adsorption (“physisorption”), — “chemical” adsorption (“chemisorption”). • Diffusion on the surface. • Chemical reaction on the surface. • Desorption. • Diffusion away from surface. 10.2.2 Physisorption and chemisorption We distinguish between “physical” adsorption (“physisorption”) and “chemical” adsorption (“chemisorption”) on a surface based upon the adsorption energies (surface binding energies). ⇒ Fig. 10.3. • Physisorption: — van-der-Waals type interaction, — typical bond energy: ∆ ≤ −40 kJ mol−1 ,
10.2 Kinetics of heterogeneous reactions (surface reactions)
219
— example: Ar on a metal or oxide surface. • Chemisorption: — formation of a chemical bond between adsorbate and surface. — typical bond energy: ∆ ≤ −100 − 300 kJ mol−1 , — example: CO on Pd: O || C | − Pd − Pd −
(10.35)
∗ chemical bond by overlap of orbital of the C atom with free bands of Pd, ∗ back donation from occupied bands into ∗ orbital of the CO substrate. ∗ Experiment shows that (Pd − C = O) ≈ 236 cm−1 .
I
Figure 10.3: Potential curves for physisorption and chemisorption.
10.2.3 The Langmuir adsorption isotherm I
Reaction scheme: A |
A() + − S − |
1
À
−1
|
−S−
− S− = “active surface site”
(10.36) (10.37)
10.2 Kinetics of heterogeneous reactions (surface reactions)
I
220
Fraction of occupied active surface sites Θ: • Adsorption rate:
1 × (1 − Θ) ×
(10.38)
−1 × Θ
(10.39)
1 (1 − Θ) = −1 Θ
(10.40)
1 = (−1 + 1 ) Θ
(10.41)
1 = −1 + 1 1 +
(10.42)
• Desorption rate: • Steady state: y y I
Langmuir isotherm (Fig. 10.4): Θ= with =
I
Figure 10.4: Langmuir adsorption isotherm.
1 −1
(10.43)
10.2 Kinetics of heterogeneous reactions (surface reactions)
I
∞ Determination of V , V , and the adsorption enthalpy ∆H : At a series ( ) as function of by of temperatures , we need to measure Θ ( ) = ∞ ( ) pressure/volume measurements in an ultrahigh vacuum apparatus.
• From Eq. 10.42:
1 1 ∞ = = 1 + Θ
• Multiplication of l.h.s. and r.h.s. with • A plot of
=
∞
(10.44)
:
1 + ∞ = + ∞
(10.45)
vs. gives a straight line (see the inset in Fig. 10.4) with slope
and intercept • • van’t Hoff plot:
I
221
∞ −1 = ( )
(10.46)
∞ −1 ) = (
(10.47)
=
slope = intercept
ln ∆ =− (1 )
(10.48)
(10.49)
Desorption rate: If ∆0 0 we can describe the desorption rate by TST (see Figs. 10.3 und 10.6). µ ¶ ‡ ∆0 −1 = exp − (10.50)
The ’s are partition functions per unit area.
10.2 Kinetics of heterogeneous reactions (surface reactions)
222
10.2.4 Surface Diffusion
I
Figure 10.5: Surface Diffusion.
I
Diffusion is an activated process: ¶ µ ∆ = 0 exp −
(10.51)
∆ = energy barrier of jump of adsorbate from one site to another. I
I
Fick’s 2nd law:
( ) 2 ( ) =× 2
(10.52)
Einstein’s diffusion law: • 1D:
∆2 = 2
(10.53)
• 2D:
∆2 = 4
(10.54)
10.2 Kinetics of heterogeneous reactions (surface reactions)
223
10.2.5 Types of heterogeneous reactions
a) Dissociative Adsorption I
Scheme: |
|
A2 () + − S − S − I
À
−2
A
|
|
−S − S−
(10.55)
Adsorption isotherm: • Adsorption rate: • Desorption rate: y Θ= with
2 2 (1 − Θ)2
(10.56)
−2 Θ2
(10.57)
(2 × )12
(10.58)
1 + (2 × )12 =
I
2
A
2 −2
Figure 10.6: Potential energy curves for dissociative adsorption.
(10.59)
10.2 Kinetics of heterogeneous reactions (surface reactions)
224
b) Surface catalyzed unimolecular dissociation I
Scheme: A |
A() + − S −
|
3
À
→4
− S−
−3
Products
I
Example: Decomposition of PH3 on a W metal surface.
I
Decomposition rate and two limiting cases: [P] A = 4 Θ = 4 × 1 + A
(10.60)
(10.61)
(1) A ¿ 1: The reaction becomes first-order in A . [P] = 4 A
(10.62)
(2) A À 1: The reaction becomes zeroth order, i.e., independent of A ! [P] = 4
(10.63)
c) Bimolecular reactions I
Scheme: A |
A() + − S −
À
−
|
−S−
(10.64)
B |
B() + − S − A
B
|
|
− S −+− S − I
À
−
|
−S−
(10.65)
A−B 5
→
|
− S−S −
→6
Products
(10.66)
Reaction rate: Assuming that A and B compete for the same active sites, we have
y
(1 − Θ − Θ ) = − Θ
(10.67)
(1 − Θ − Θ ) = − Θ
(10.68)
5 [P] = 5 Θ Θ = (1 + + )2
(10.69)
10.2 Kinetics of heterogeneous reactions (surface reactions)
I
225
Limiting cases: It is often the case that one species is adsorbed much more strongly than the other. • Assume, for instance, that ¿ . Then, [P] 5 = (1 + )2
(10.70)
(1) À 1:
[P] 5 (10.71) = The reaction is inhibited by B, which is strongly adsorbed: At high , all active sites are blocked by B.
(2) ¿ 1:
[P] = 5 The reaction is truly second order.
(10.72)
[P] first ⇒ We can see from these two cases that, as increases, the rate increases, reaches a maximum, and then decreases again.
d) Example 1: Heterogeneous oxidation of CO I
Figure 10.7: Heterogeneous oxidation of CO on Pt. (a) Langmuir-Hinshelwood and Eley-Rideal mechanisms, (b) potential energy diagram (values in kJ mol−1 ).
10.2 Kinetics of heterogeneous reactions (surface reactions)
226
e) Example 2: Haber-Bosch NH3 synthesis
I
Figure 10.8: Potential energy diagram for the catalytic reaction N2 + 3H2 → 2 NH3 on a Fe3 O4 catalyst according to Ertl and co-workers (1982). Energies in kJ mol−1 .
f) How can we control and accelerate heterogeneous reactions? The answer depends on which step is rate determining: • If diffusion to/from the surface is the rate determining step, we can accelerate the reaction by stirring • If the adsorption step is rate determining (because of an activation energy), we can accelerate the reaction by increasing and and, in particular, by increasing surface area.
10.2 Kinetics of heterogeneous reactions (surface reactions)
227
11. Reactions in solution
“Everyone has problems - chemists have solutions.”
I
Comparison of gas and liquid phase reactions (@ T = 298 K and p = 1 bar): • Molecules in the gas phase occupy ≈ 02 % of the total volume, while molecules in the liquid phase occupy ≈ 50 % of the total volume. • Molecules in liquid are in permanent contact with each other, and undergo permanent collisions with each other. • The crossing of a potential barrier in a reaction is not continuous as in gas phase but affected by multiple collisions during the crossing. • As a result of the permanent collisions, we can expect the reactant molecules to be in thermodynamic equilibrium even above 0 . Thus TST should be applicable. • However, it is very difficult to evaluate the partition functions in liquids. Hence, one usually employs the thermodynamic version TST. • Solvent molecules act as an efficient heat bath.
I
Observations on liquid phase reactions: Experience has shown that reactions in liquids have similar rates as in the gas phase, except if • the reaction occurs with the solvent, • there are strong interactions of the reactant molecules with the solvent (e.g., ionic reactions; ⇒ solvent shells), • the reactions are diffusion controlled.
11.1 Qualitative model of liquid phase reactions In contrast to gas phase reactions for which we have advanced theories (⇒ transition state theory, unimolecular rate theory), the theory for liquid phase reactions is much less well developed. However, we can understand the required basic reaction steps as sketched in Fig. 11.1 and a number of limiting cases resulting from these steps.
11.1 Qualitative model of liquid phase reactions
228
I
Figure 11.1: Structure and dynamics of solutions.
I
Formal kinetics: We distinguish between the formation of the encounter complex in the solvent cage (by diffusion) and the actual reaction: +
À
−
y
y
→
[{}] = [] [] − − [{}] − [{}] ≈ 0 [{}] = [] [] − + [ ] [] [] = [] [] = [{}] = − +
with =
I
{}
− +
(11.1)
(11.2) (11.3)
(11.4)
(11.5)
Rate constant for liquid phase reactions: =
− +
(11.6)
11.1 Qualitative model of liquid phase reactions
I
229
Two limiting cases: (1) Diffusion control: À − y =
(11.7)
This case is observed when = 0 and the diffusion coefficient is small. We shall find below that = 4 (11.8) (2) Reaction (or activation) control: ¿ − y =
= {} −
(11.9)
In this case, the reaction is said to be activation controlled because usually it has a sizable . can be described by TST: =
∆ + −∆ +
(11.10)
11.2 Diffusion-controlled reactions
230
11.2 Diffusion-controlled reactions 11.2.1 Experimental observations The transition to diffusion control can be observed in the gas phase by studying the kinetics in highly compressed gases (up to = 1000 bar) or, even better, in supercritical media. At ≈ 1000 bar, the gas density reaches the density of the liquid phase. I
Smoluchowsky-Kramers limit: At high densities, the rate constant becomes ∝∝
I
1
(11.11)
Figure 11.2: Diffusion control of a unimolecular reaction: At very high pressures, the fragments recombine before they become separated by diffusion y the reaction becomes diffusion controlled.
11.2.2 Derivation of k In order to derive a theoretical expression for , we consider the distribution of the molecules around a molecule in the center of a sphere as shown in Fig. 11.3.
11.2 Diffusion-controlled reactions
I
231
Figure 11.3: Radial distribution of reactants in a diffusion-controlled reaction.
(1) We start by considering the boundary conditions for the concentration [] (cf. Fig. 11.3): • At a distance → ∞, the concentration of is identical to the mean concentration in the solution, i.e., → ∞ y [] = []=∞ = []
(11.12)
• At the distance = for a diffusion controlled reaction, the concentration of vanishes because of its fast reaction with , i.e., → 0 y []=0 = 0
(11.13)
[] (2) The result is a concentration gradient around , which gives rise to a net ← − flux by diffusion as described by Fick’s first law: = × ×
[]
(11.14)
We define this flux, measured in units of molecules/s, in the direction from = ∞ (high concentration) to = (low concentration); thus there is no − sign in the usually written form of Fick’s law. is the mean diffusion coefficient, measured in m2 s, = +
(11.15)
11.2 Diffusion-controlled reactions
232
(3) With = 42 as the surface of the sphere with radius around , we obtain = × 42 ×
[]
(4) To determine the concentration profile [] , we integrate over [] : Z ∞ Z []∞ [] = 2 4 [] y
[]|∞ y
¯∞ ¯ ¯ =− 4 ¯
[] − [] =
−0 4
(11.16)
(11.17)
(11.18) (11.19)
(5) is found from the boundary conditions that [] = 0 at = : = 4 × []
(11.20)
(6) The concentration gradient of [] is obtained by inserting the above result for into Eq. 11.19: [] [] − [] = (11.21) y ³ ´ [] = [] 1 − (11.22) This expression is plotted in Fig. 11.3. (7) To determine , we write the rate of the diffusion controlled reaction as [ ] = [] Inserting the expression for from Eq. 11.20, we obtain [ ] = 4 × [] []
(11.23)
(11.24)
The diffusion-limited rate constant is thus (in molecular units) = 4
(11.25)
(8) Nernst-Einstein relation between diffusion constant and viscosity for spherical particles with radius : = (11.26) 6 y ∝ (11.27) (9) Keep in mind that diffusion is an activated process: = 0 − Typical value for H2 O: ≈ 15 kJ mol−1 .
(11.28)
11.2 Diffusion-controlled reactions
I
Typical values:
y Typical magnitude of :
I
≈ 2 × 10−5 cm2 s−1 ≈ 5 × 10−8 cm
(11.29) (11.30)
≈ 8 × 109 l mol−1 s−1
(11.31)
Table 11.1: Diffusion constants of ions in aqueous solution at = 298 K. ion H+ Li+ Na+
I
233
( cm2 s−1 ) 91 × 10−5 10 × 10−5 13 × 10−5
ion OH− Cl− Br−
( cm2 s−1 ) 52 × 10−5 20 × 10−5 21 × 10−5
Examples of diffusion controlled reaction in solution: • radical recombination reactions, e.g. + → 2 :
= 13 × 1010 l mol−1 s−1
(11.32)
• electronic deactivation (quenching) in solution • H+ transfer reactions
+ + − → 2 :
= 14 × 1011 l mol−1 s−1
(11.33)
This reaction is ten times faster than a typical diffusion controlled reaction, because H+ transfer is a concerted process (Grotthus mechanism, see Fig. 11.4)! I
Figure 11.4: Structures involved in fast H+ transfer processes in liquid water.
11.3 Activation controlled reactions
234
11.3 Activation controlled reactions The rate constant for an activation controlled reaction is = {} I
(11.34)
Estimation of K{} : We can estimate the equilibrium constant {} using simple arguments based on the coordination number. Accordingly, the concentration of the encounter complex is [{}] = [] × (probability of finding next to ) [] = [] × []
(11.35) (11.36)
where [] is the solvent concentration and is the coordination number. y {} =
[{}] = [] [] []
(11.37)
For H2 O at = 298 K, [] = 555 mol l−1 . Thus, taking for example = 8, we find that {} = 014 l mol−1 (11.38) I
Application of thermodynamic TST: TST can be applied in two ways: (1) Considering the overall reaction, we can write =
¡ ¢ exp −∆‡
(11.39)
where ∆‡ is the overall Gibbs free enthalphy for the reaction. (2) We can also separate the effects from and {} by writing ³ ´ ª {} = exp −∆{} and
(11.40)
¡ ¢ exp −∆+ (11.41) ∆ª {} is the Gibbs free enthalpy change for the formation of the encounter complex, and ∆+ is the Gibbs free enthalphy of activation from the encounter complex. Thus, ³ ³ ´ ´ + exp − ∆ª (11.42) + ∆ = {} =
I
Pressure dependence of the rate constant: ⇒ See section 7.3.5.
11.4 Electron transfer reactions (Marcus theory)
235
11.4 Electron transfer reactions (Marcus theory) Electron transfer is what happens in oxidation-reduction reactions. Hence, electron transfer reactions are extremely important. They are strongly affected by the solvent because the change of the charge distribution results in a large reorganization energy. I
Figure 11.5:
I
Marcus theory: For a simplified derivation of ( ) we refer to Fig. 11.5. (1) The energy of the donor and acceptor molecules will generally depend on all 3 −6 internal coordinates. We consider the dependence on an effective coordinate , which describes the minimum energy path from reactant to product (Fig. 11.5). (2) The exergonicity of the ET reaction ∆ª is negative for an exergonic reaction (convention; cf. Fig. 11.5). (3) For the donor (D) and the acceptor (A), the Gibbs energy profiles () and () shall be of parabolic form, which we may write as
and
() = 2
(11.43)
() = ( − )2 + ∆ª .
(11.44)
The origin = 0 has been placed at the minimum of the donor; the acceptor has its minimum at = .50 50
We assume that ∝ 2 , taking any proportionality constant into account by proper rescaling of the abscissa.
11.4 Electron transfer reactions (Marcus theory)
236
(4) The ET rate constant is described by TST as =
¡ ¢ exp −∆+
(11.45)
The Gibbs free energy of activation
2 ∆+ =
(11.46)
is the difference from the donor minimum to the TS at the donor/acceptor curve crossing at point . (5) In order to relate ∆+ to the potential curves, we define the “reorganization energy” as the energy that the acceptor would have at the equilibrium geometry of the donor. This is the energy of the acceptor parabola () at = 0. If we count from the minimum of the acceptor parabola (), we have = (− )2 = 2
(11.47)
(6) We now determine the energy of the intersection point of the two parabolas from the condition ( ) = ( ) . (11.48) Inserting into Eqs. 11.43 and 11.44 yields, and using Eq. 11.47, yields 2 = ( − )2 + ∆ª 2 − 2 + 2 + ∆ª = 2 − 2 + + ∆ª =
(11.49) (11.50) (11.51)
2 = + ƻ
(11.52)
y y
= y +
∆ = y
2
+ ƻ 2
( + ƻ )2 ( + ƻ )2 = = 4 2 4
à ! ( + ∆ª )2 exp − ( ) = 4
(11.53)
(11.54)
(11.55)
(7) Limiting cases: We can distinguish between • −∆ª : Regular region – the reaction becomes faster, the more exergonic the reaction becomes. • −∆ª = : turning point
• −∆ª : Inverted region – the reaction becomes slower, the more exergonic the reaction becomes.
11.4 Electron transfer reactions (Marcus theory)
237
I
Conclusion: Equation 11.55 predicts the ET rate constant to increase with increasing exoergicity (−∆ª ) of the reaction. However, once −∆ª , the rate constant is predicted to decrease again (see Figs. 11.6D and 11.7). The prediction of this unexpected inverted Marcus regime before any experimental data were available is the hallmark of Marcus theory.
I
Figure 11.6:
I
Figure 11.7:
11.5 Reactions of ions in solutions
238
11.5 Reactions of ions in solutions Reactions of ionic species in liquids are one exception to the rule that reactions in the liquid phase usually have similar rates as in the gas phase (the other exception being electron transfer reactions). The reason is that the charged species are very sensitive to their environment, in particular solvation. Changes of the charge distribution are accompanied by correspondingly large solvation shell rearrangements.
11.5.1 Effect of the ionic strength of the solution The main effect arises from the stabilization of shielding every ion in solution by the oppositely charged ”ionic atmosphere” or “ion cloud” (⇒ Debye-Hückel theory, Appendix ??). I
h i ‡ Equilibrium constant for AB :
£ ¤ ‡ ‡ ‡ = = [] [] ‡
I
(11.56)
Activity coefficient from Debye-Hückel theory: log = − 2 12
(11.57)
with (for water at = 298 K, according to Debye-Hückel theory) = 0509 l12 mol−12
I
Ionic strength: =
1X 2 2
(11.58)
(11.59)
11.5.2 Kinetic salt effect I
TST rate constant: £ ¤ [ ] = ‡ ‡ = ‡ ‡ ‡ [] []
(11.60)
Defining 0 as the rate constant for the ideal solution, where all = 1, this becomes = 0
‡
(11.61)
11.5 Reactions of ions in solutions
I
239
Dependence of the rate constant on the ionic strength: ´ ³ ‡ 2 12 log = log 0 − 2 + 2 −
(11.62)
‡ = + . with
‡ = + , we obtain Inserting
¡ ¢ log = log 0 − 2 + 2 − ( + ) 2 12
y
log
I
12 = +2 0
(11.63)
(11.64)
Conclusions: • If and have the same sign, then will increase with increasing , because the repulsion is reduced by the shielding effect of the ion cloud. • If and have opposite sign, then will decrease with increasing , because the attraction is reduced by the shielding effect of the ion cloud.
I
Figure 11.8: Kinetic salt effect for reactions of ions in solution.
11.5 Reactions of ions in solutions
11.6 References Eigen 1954a M. Eigen, Disc. Faraday Soc. 17, 194 (1954). Eigen 1954b M. Eigen Z. Physik. Chem. NF 1, 176 (1954).
240
11.6 References
241
12. Photochemical reactions 12.1 Radiative processes in a two-level system; Einstein coefficients I
Three fundamental radiative processes (Einstein): • absorption, • spontaneous emission, • stimulated emission:
I
Rate equation using Einstein coefficients: 2 1 =− = 12 () 1 − 21 () 2 − 21 2
(12.1)
with
8 3 21 3 and (for a true two-level system, with 1 = 2 ) 21 =
12 = 21
I
Figure 12.1: Absorption, stimulated emission, and spontaneous emission.
(12.2)
(12.3)
12.2 Light amplification by stimulated emission of radiation (LASER)
12.2 Light amplification by stimulated emission of radiation (LASER) I
Requirements for a laser: • Optical medium with population inversion, • pump source to achieve population inversion, • optical resonator for feedback.
I
Figure 12.2: Light amplification by stimulated emission of radiation (LASER).
I
Figure 12.3: Three-level and four-level laser.
242
12.3 Fluorescence quenching (Stern-Volmer equation)
243
12.3 Fluorescence quenching (Stern-Volmer equation) I
Kinetic model:
A + → A∗ A∗ → A + ∗ A +Q →A +Q
1
(12.4)
If any other radiationless processes can be neglected, we find [A∗ ] =
I
1 [A] + [Q]
Fluorescence intensity I0 in the absence of the quencher Q: 0 ∝ [A∗ ] = 1 [A]
I
(12.6)
Fluorescence intensity I in the presence of the quencher Q: ∝ [A∗ ] =
I
(12.5)
1 [A] + [Q]
(12.7)
Stern-Volmer equation: 0 + [Q] + [Q] = = 1 [A] × 1 [A]
(12.8)
0 = 1 + 0 [Q]
(12.9)
y
with the I
Radiative lifetime: 0 =
I
1
(12.10)
Figure 12.4: Stern-Volmer plot. A plot of 0 vs. [Q] should give a straight line with intercept 1 and slope 0 [Q].
12.4 The radiative lifetime 0 and the fluorescence quantum yield Φ
244
12.4 The radiative lifetime τ 0 and the fluorescence quantum yield Φ I
Determination of the radiative lifetime τ 0 : Assuming that there are no competing nonradiative processes, we can determine the radiative lifetime 0 by (1) time-resolved measurements of the (exponential) fluorescence decay after pulsed excitation, (2) high-resolution measurement of the natural (Lorentzian) linewidth according to ∆ ∆ = ~
(12.11)
1 2 ∆ 0
(12.12)
y 0 =
(3) evaluation from the molar absorption coefficient (Lambert-Beer) using the relationship between the Einstein coefficients 21 and 21 ,51 21 =
8 3 21 3
(4) calculation from first principles of the value of ¯ ¯2 ¯ ¯ ∝ 21 ∝ 21 ∝ 12 ∝ . I
(12.13)
(12.14)
Fluorescence quantum yield: Apart from quenching, the fluorescence is reduced by other radiationless processes (IC, ISC, R, . . . ). The experimentally measured fluorescence lifetime is therefore smaller (often much smaller) than the radiative lifetime 0 . The ratio is the fluorescence quantum yield: Φ =
51
+
= + + + [Q] + 0
(12.15)
The actual procedure, which involves integration over the absorption band, corrects for the influence of the refraction index of the solvent, etc., is called the Strickler-Berg analysis.
12.5 Radiationless processes in photoexcited molecules
245
12.5 Radiationless processes in photoexcited molecules
I
Figure 12.5: Jablonski Diagram.
I
Primary photochemical processes: ¯ ¯2 • Absorption of a photon (depends on transition dipole moment ¯ ¯ ),
• Intramolecular processes:
— fluorescence (≡ spontaneous emission of a photon, FLUO), — stimulated emission (SE),
(⇒ Einstein coefficients)
— intramolecular vibrational (energy) redistribution (IVR), — internal conversion (IC), — chemical reaction in the excited state (RXN), — intersystem crossing (ISC), — phosphorescence (PHOS). • Intermolecular (i.e., collisional) processes: — electronic deactivation (quenching, Q), ∗ may affect fluorescence as well as phosphorescence,
— vibrational energy transfer (VET), — rotational energy transfer (RET),
— collision-induced intersystem crossing (CIISC).
12.5 Radiationless processes in photoexcited molecules
246
12.5.1 The Born-Oppenheimer approximation and its breakdown
a) The full Hamiltonian We consider a non-moving, non-rotating molecule in the laboratory framework. The molecule is described by the Schrödinger equation (SE) ˆ (r R) = (r R)
(12.16)
´ ³ ˆ − (r R) = 0
(12.17)
or
with being the energy eigenvalue associated with the wavefunction (r R) as function of the electronic (e) coordinates r and the nuclear (N) coordinates R. ˆ appearing in the SE can be written as The Hamilton operator ˆ = ˆ + ˆ = ˆ (r) + ˆ (R) + (r R)
(12.18)
where ~2 X ˆ = − ∇2 2 =1
~2 X ˆ ∇2 = − 2 =1
(12.19) (12.20)
and (r R) = (R) + (r R) + (r)
(12.21)
with = +
0 2 X X 40 0 0 =1 0
X 2 X = − 40 =1 =1
2 X X 1 = + 40 0 0 =1 0
(12.22)
(12.23) (12.24)
Electronic spin, spin-orbit interaction, and nuclear spins are neglected here. Further, we have left aside the so-called electronic mass and electronic nuclear cross polarization terms, which appear in the case of a moving molecule as result of the separation of the center-of-mass. b) Adiabatic separation of nuclear and electronic motion We now want to separate the slow motion of the nuclei from the the fast motion of the electrons. This separation can be made because the fast electrons can be assumed
12.5 Radiationless processes in photoexcited molecules
247
to very rapidly follow the slow nuclei. The kinetic energy of the nuclei is assumed to be small compared to the electronic energy. Thus, for every nuclear configuration R, there is an electronic wavefunction el (r R) belonging to the electronic state |i specified by quantum number . This el (r R) depends on the nuclear coordinates R, but only very little on the nuclear velocities. We say that the electrons adiabatically follow the periodic vibrational motion of the nuclei. This approximation is therefore called adiabatic approximation. I
ˆ Ansatz for H: ˆ0 ˆ = ˆ0 +
(12.25)
with 2 X ˆ 0 = ˆ + (r R) = − ~ ∇2 + (r R) 2 =1 ~2 X 0 ˆ ˆ = = − ∇2 2 =1
I
(12.26) (12.27)
ˆ 0: The zero-order Hamiltonian H ˆ 0 depends only on the electron kinetic energy op• The zero-order Hamiltonian erator and the potential energy and describes the rigid molecule (all R are fixed!) via the unperturbed SE ˆ 0 el (r; R) = (0) (R) el (r; R)
(12.28)
• The zero-order electronic wave functions el (r; R) describe the electrons and ¯ ¯2 their distribution ¯el (r; R)¯ for the case of fixed nuclei in the electronic state |i with the electronic energy el . • The el (r; R) depend on R only as a parameter, not as a variable (we do not differentiate or integrate with respect to the in the zero-order SE). • The el (r; R) form a complete orthonormal system. • The eigenvalues (0) (R) of the electronic SE are what we have come to know as the molecular potential energy functions (or hypersurfaces) (R) of the molecule with all nuclei at rest.
I
ˆ 0: The perturbation H ˆ 0 is the nuclear kinetic energy operator ˆ . • The perturbation
12.5 Radiationless processes in photoexcited molecules
248
c) Solution of the complete SE • To solve the complete SE
we use the ansatz
³ ´ ˆ0 + ˆ 0 − (r R) = 0
(r R) =
X
(R) el (r; R)
(12.29)
(12.30)
with the expansion coefficients = (R) depending only on R but not on r. • Inserting the ansatz 12.30 into the complete SE, ³ ´X 0 0 ˆ ˆ + − (R) el (r; R) = 0
(12.31)
and doing some algebraic manipulation, we arrive at a
• system of coupled equations for the electronic wavefunctions el (r; R) for the electronic state el (r; R) and the nuclear wavefunctions (R): ˆ 0 el (r; R) = (0) (R) el (r; R)
(12.32)
and ˆ 0 (R) +
X
¡ ¢ (R) (R) = − (0) (R) (R)
(12.33)
with the coupling coefficients Z ˆ 0 (r; R) r (R) = ∗ (r; R) r
2
−~
Z
∗ (r; R)
r
X
1 (r; R) r
(12.34)
• We interprete these Eqs. as follows: (0)
(1) (R) can be interpreted as the zero-order potential functions of the th zero-order electronic state that is given by the solution of the electronic SE (Eq. 12.28) for fixed nuclei R. (2) Without the (R), Eq. 12.33 describes the kinetic energy of the nuclei (0) in the potential (R).52 Eq. 12.33 is thus simply the vibrational SE for (0) the electronic potential (R). (3) The coefficients (R) defined by Eq. 12.34 are coupling terms that connect (i.e., mix) the electronic states and . They describe how the different electronic states and are coupled by the nuclear motion (the two terms on the RHS resulting from the nuclear kinetic energy operator ˆ ; see Eq. 12.34).
52
We usually denote these potential energy functions as (R).
12.5 Radiationless processes in photoexcited molecules
249
d) Born-Oppenheimer approximation The Born-Oppenheimer (BO) approximation completely neglects the coupling coefficients . The complete SE is therefore reduced to two uncoupled equations, which we denote as electronic SE ˆ 0 el (r) = (0) (R) el (r) and nuclear (vibrational) SE h i (0) ˆ + (R) = (R)
(12.35)
(12.36)
which describe, respectively, the electronic wavefunction el for fixed nuclear coordinates R in the electronic state el and the set of nuclear wavefunctions (R) for the energy state of the nuclei in the electronic state el . e) Breakdown of the Born-Oppenheimer approximnation When two (or more) electronic states el (R), el (R) become nearly isoenergetic or, at a crossing, even degenerate, the coupling matrix element (R) may no longer be neglected, because through the (R), the nuclear motion leads to a mixing between the different electronic BO-states. I
This breakdown of the BO approximnation leads to non-adiabatic coupling of electronic states: • For a diatomic molecule, the coupling leads to an avoided crossing of the two electronic potential curves. • For polyatomic molecules, the coupling can lead to a conical intersection (CI) between the two electronic potential energy hypersurfaces. As will be outlined in the following, the existence of conical intersections between different excited electronic potential energy hypersurfaces and between an excited and the ground electronic potential energy hypersurface has dramatic consequences for the dynamics of photochemical reactions.53
12.5.2 Avoided crossings of two potential curves of a diatomic molecule I
Illustration of an avoided crossing using two diabatic model potential functions + coupling term: 1 () = 1 [1 − exp [− 1 ( − 1 )]]2 2 () = 2 + exp [− 2 ( − 2 )] 12 () = 1 [1 + tanh [−2 ( − 12 )]]
(12.37) (12.38) (12.39)
12 () is just a convenient model function with the physically correct behavior 12 () → 0 for → ∞. 53
The recognition since about the years 2000 - 2005 that conical intersections in polyatomic molecules are the rule rather than the exception has indeed revolutionized our understanding of photochemical reactions.
12.5 Radiationless processes in photoexcited molecules
I
Figure 12.6: Diabatic model potential curves for a diatomic molecule.
I
Figure 12.7: Avoided crossing of two potential curves of a diatomic molecule.
250
12.5 Radiationless processes in photoexcited molecules
I
251
Figure 12.8: Conical intersection of potential energy surfaces of a polyatomic molecule.
12.5.3 Quantum mechanical treatment of a coupled 2×2 system We can easily verify the “avoided crossing” of two coupled potential energy curves as follows: I
Schrödinger equation:
I
Hamilton-Operator:
I
Ansatz:
( − ) |i = 0
(12.40)
= (0) + (1)
(12.41)
|i = 1 |1 i + 2 |2 i
(12.42) (0)
with orthonormal basis vectors that are eigenvectors of , i.e., ½ ® 1 for = | = = 0 for 6=
(12.43)
and the zero-order energies
(0)
(12.44)
(0)
(12.45)
(0) |1 i = 1 |1 i
(0) |2 i = 2 |2 i
12.5 Radiationless processes in photoexcited molecules
I
252
Solution of the Schrödinger equation: Insertion of the ansatz (Eq. 12.42) into the SE and multiplication from the left by either h1 | or by h2 | gives the two equations: 1 h1 | ( − ) |1 i + 2 h1 | ( − ) |2 i = 0 1 h2 | ( − ) |1 i + 2 h2 | ( − ) |2 i = 0
(12.46) (12.47)
This is a system of coupled linear equations (“secular equations”): 1 (11 − ) + 2 12 = 0 1 21 + 2 (22 − ) = 0
(12.48) (12.49)
with the “matrix elements” (integrals over all r resp. R): = h | | i = h | (0) + (1) | i = h | (0) | i + h | (1) | i
(12.50) (12.51) (12.52)
Specifically, we have (0)
(12.53)
(0) 2
(12.54)
11 = h1 | (0) |1 i = 1 22 = h2 |
(0)
|2 i =
and 12 = h1 | (1) |2 i = h2 | (1) |1 i = 21
(12.55)
Therefore, we obtain the secular equations as follows: I
I
I
Secular equations: ³ ´ (0) 1 1 − + 2 12 = 0 ³ ´ (0) 1 21 + 2 2 − = 0
(12.56) (12.57)
Secular determinant: A non-trivial solution for the secular equations requires that ¯ ¯ ¯ (0) ¯ ¯ 1 − 12 ¯ (12.58) ¯ ¯=0 ¯ 21 2(0) − ¯ Eigenvalues: ³ ´³ ´ (0) (0) 0 = 1 − 2 − − |12 |2 ´ ³ (0) (0) (0) (0) = 1 2 − 1 + 2 + 2 − |12 |2
with because
(1)
is Hermitian.
|12 |2 = 12 21
(12.59) (12.60)
(12.61)
12.5 Radiationless processes in photoexcited molecules
With some rewriting54 , we obtain ´ 1 r³ ´2 1 ³ (0) (0) (0) (0) (0) (0) 1 + 2 ± 1 + 2 − 41 2 + 4 |12 |2 ± = 2 2
y
I
´ 1 1 ³ (0) (0) 1 + 2 ± ± = 2 2
r³ ´2 (0) (0) 1 − 2 + 4 |12 |2
253
(12.62)
(12.63)
Shorthand form: Using the abbreviations ± (R) = ±
(12.64)
1 (R) =
(12.65)
(0) 1 (0) 2
2 (R) = 2 (R) = |12 |2 1 (1 (R) + 2 (R)) 2 1 ∆ (R) = (1 (R) − 2 (R)) 2 Σ (R) =
(12.66) (12.67)
(12.68) (12.69)
we can rewrite the solution for the eigenvalues + and − in a compact form as q ± (R) = Σ (R) ± (∆ (R))2 + ( (R))2
(12.70)
12.5.4 Radiationless processes in polyatomic molecules: Conical intersections I
“Traditional” view on the cis-trans isomerization of C2 H4 : • Reaction coordinate can be approximated by torsion angle (0 ≤ ≤ 180◦ ). • Reaction proceeds via a perpendicular transition state ( = 90◦ ), in which the C atoms are connected by a single bond. • Correlation diagram shows that the orbital of the trans-isomer transforms into the ∗ orbital of the cis-isomer and vice versa. • Crossing of the two diabatic states corresponds to an avoided crossing of the adiabatic states. 54
³ ³ ´2 ´2 ³ ´2 (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) − 41 2 = 1 + 2 + 21 2 − 41 2 1 + 2 ³ ³ ´2 ³ ´2 ´2 (0) (0) (0) (0) (0) (0) = 1 + 2 − 21 2 = 1 − 2
12.5 Radiationless processes in photoexcited molecules
254
• The reaction is thus thermally forbidden because of the high energy barrier at the avoided crossing between the trans and cis isomers on the left and right. • The reaction is photochemically allowed because of the barrierless adiabatic correlation between the ∗ states on the left and right. I
Figure 12.9: Cis-trans isomerization of C2 H4 : HMO picture with transformation of and ∗ orbitals and conical intersection.
I
Figure 12.10: Internal coordinates relevant for the cis-trans isomerization of C2 H4 . The out-of-plane bending angle plays a role as coupling mode leading to a CI; the angles and play a role in the isomerization to H3 CCH (also by a CI).
12.5 Radiationless processes in photoexcited molecules
255
I
Figure 12.11: Diabatic model PES’s for C2 H4 .
I
Figure 12.12: Adiabatic model PES’s for C2 H4 with the conical intersection at = 90◦ .
12.5 Radiationless processes in photoexcited molecules
I
Figure 12.13: Wave packet motion in a complex photo-induced reaction.
12.5.5 Femtosecond Time-Resolved Experiments
I
Figure 12.14:
256
12.5 Radiationless processes in photoexcited molecules
I
Figure 12.15:
I
Figure 12.16:
257
12.5 Radiationless processes in photoexcited molecules
I
Figure 12.17:
I
Figure 12.18:
258
12.5 Radiationless processes in photoexcited molecules
13. Atmospheric chemistry* (skipped for lack of time in the semester)
259
13 Atmospheric chemistry*
14. Combustion chemistry* (skipped for lack of time in the semester)
260
14 Combustion chemistry*
15. Astrochemistry* (skipped for lack of time in the semester)
261
15 Astrochemistry*
16. Energy transfer processes* (See lecture in M.Sc. course.)
262
16 Energy transfer processes*
17. Reaction dynamics* (See lecture in M.Sc. course.)
263
17 Reaction dynamics*
18. Electrochemical kinetics* (skipped for lack of time in the semester)
264
Appendix B
265
Appendix A: Useful physicochemical constants
I
Table A.1: List of useful physical constants (Cohen 1986). physical constant symbol Loschmidt (or Avogadro) number gas constant Boltzmann constant electron charge Faraday constant speed of light Planck’s constant ~ electron mass proton mass neutron mass atomic mass unit e Rydberg constant Bohr radius 0
I
electric field constant
0
magnetic field constant
0
value = 60221367 × 1023 mol−1 = 8314511 J mol−1 K−1 = 1380658 × 10−23 J K−1 = 160217733 × 10−19 C = 96485309 C mol−1 = 299792458 × 108 m s−1 = 66260755 × 10−34 J s ~ = 10547266 × 10−34 J s = 91093897 × 10−31 kg = 16726231 × 10−27 kg = 16749286 × 10−27 kg = 16605402 × 10−27 kg e = 10973731534 × 107 m−1 0 = 529177249 × 10−11 m
0 = 1(0 2 ) 0 = 8854187817 × 10−12 F m−1 0 = 4 × 10−7 N A−2 0 = 125664 × 10−6 N A−2
References: Cohen 1986 E. R. Cohen and B. N. Taylor, The 1986 Adjustment of the Fundamental Constants, CODATA Bull. 63, 1 (1986). An updated list is contained in every August issue of Physics Today.
Appendix B
266
Appendix B: The Marquardt-Levenberg non-linear least-squares fitting algorithm With the omnipresence of the computer, the Marquardt-Levenberg algorithm (Bevington 1992, Press 1992) has become an extremely important data analysis tool. It is generally the method of choice in cases where it is not possible to make simple linear plots to determine rate constants. I
Problem: Experimentally, we measure the values of some quantity at the points , , . Suppose we take measurements. Now suppose we would like to describe our data points by some model. Towards these ends, we take a physically reasonable model function = ( ; (1 )) (B.1) which we would like to fit to our data in such a way that ( ) describes the data points in “the best possible way”. The model function depends directly on the variables , and it also depends parametrically on nonlinear parameters . In kinetics, for example, is usually some concentration, e.g. , which depends on time i.e., = (). The parameters may be the rate constants (or decay times ) and the initial concentration 0 . Our job is to adjust and optimize these parameters by somehow “fitting” them. The method of choice is, of course, “least-squares fitting”: We start by defining the so-called figure-of-merit or goodness-of-fit function55 " # 2 X − ( )] [ (B.2) 2 = 2 =1 Here, the and the independent variables , are the measured quantities, the are the experimental uncertainties of the , and ( ) is the calculated value of the model fit function at the point { } (the parameters 1 are omitted here in ). What we have to do is to adjust the parameters in the model function such that 2 reaches a minimum, i.e. we have to search the -dimensional parameter space for the (global) minimum of 2 .
I
Solution: The Marquardt-Levenberg algorithm is a method to determine this minimum of 2 as function of the fit parameters in the model fit function . The minimum of 2 as function of the fit parameters a is defined by the conditions " # 2 X [ − ( )]2 =0 (B.3) = =1 2 ¸ ∙ X [ − ( )] ( ) = −2 (B.4) 2 =1 Thus, taking the partial derivatives of 2 with respect to each of the parameters , we obtain a set of coupled equations in the unknown parameters . These equations are, in general, nonlinear in the . 55
Deutsch: Fehlerquadratsumme.
Appendix B
267
The Marquardt-Levenberg algorithm gives a recipy for finding the minimum of 2 using a combination of the method of steepest descent (following the gradient of the hypersurface defined by 2 as a function of the parameters 1 and, near the minimum of 2 , a parabolic Taylor series expansion of the fitting function around the minimum). At the end of a fit, we shall usually report the (± 2 standard deviations), the standard deviation of the ’s, and the so-called reduced-2 , which is 2 divided by the number of degrees of freedom, = − , i.e., " # 2 X − ( )] [ 1 (B.5) 2 = − =1 2 I
Example 1: Exponential decay. Suppose we measure the time dependent concentration () of a molecule in some quantum state, which is populated at some time 0 by a laser pulse and then decays monoexponentially. As a model fit function, we use the expression () = 1 exp [−2 ( − 3 )] + 4 + 5 (B.6) The parameters we are interested in are 2 (the decay rate coefficient) and 1 (the initial amplitude or initial concentration). The parameter 3 just describes the trigger delay time (= 0 ). We have also added the parameters 4 and 5 to describe a constant background term (due to some offset of our experimental signal) and a linear drift in this background term (perhaps due to some experimental instability, such as a gradual temperature increase in the lab). Our figure of merit is then " # 2 X − ()] [ 2 = (B.7) 2 =1 The (in general nonlinear) conditions for the minimum of 2 , from which we determine the parameters , are " # ¸ ∙ X 2 [ − ()] () X [ − ()]2 = −2 =0 (B.8) = 2 1 1 =1 2 1 =1 2 = 2
(B.9) (B.10)
Figure B.1 below shows a simple two-parameter fit (parameters 1 and 1) to such a signal.
Appendix B
268
I
Figure B.1: Exponential decay curve of a particular vibration-rotation state of the CH3 O radical resulting from the unimolecular dissociation reaction of the radical according to CH3 O → H + H2 CO (Dertinger 1995). The small box is the output box from a fit using the ORIGIN program.
I
Example 2: Multiexponential decay. In femtosecond spectroscopy, we often observe ultrafast multiexponential decays of laser-excited molecules. The laser prepares an excited wavepacket, which usually does not decay single-exponentially. Further, we need to take into account the final duration of the pump laser pulse (by deconvolution, or by forward convolution). • Model function to be fitted to measured data: X () = exp (− )
(B.11)
with adjustable parameters and .
• Instrument response function (IRF): Often represented by a Gaussian centered at time 0 ! Ã 1 ( − 0 )2 √ exp − (B.12) () = 2 2IRF IRF 2 with width parameter IRF related to the full width at half maximum (FWHM) of the IRF by √ (B.13) FWHM = 8 ln 2 ≈ 2355IRF
Appendix C
269
• Convolution of the molecular intensity () and () gives the signal function Z+∞ Z+∞ 0 0 0 () = ( ) ( − ) = ( − 0 ) (0 ) 0 = () ⊗ () (B.14) −∞
−∞
where ⊗ denotes the convolution. • Resulting model function to be fitted to the data: ¸∙ µ ¶¸ ∙ 2 ( − 0 ) − 2IRF 1 IRF ( − 0 ) 1X √ 1 + erf + exp − () = 2 2 2 2 IRF
(B.15) where erf () is the error function and is a simple constant background term (can be replaced by background + drift + ).
I
Figure B.2: Excited-state relaxation dynamics of the adenine dinucleotide after UV photoexcitation.
I
References: Bevington 1992 P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, Boston, 1992. Dertinger 1995 S. Dertinger, A. Geers, J. Kappert, F. Temps, J. W. Wiebrecht, Rotation-Vibration State Resolved Unimolecular Dynamics of Highly Vibrationally Excited CH3 O (2 ): III. State Specific Dissociation Rates from Spectroscopic Line Profiles and Time Resolved Measurements, Faraday Discuss. Roy. Soc. 102, 31 (1995). Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1992. Versions are also available for C and Pascal.
Appendix C
270
Appendix C: Solution of inhomogeneous differential equations C.1 General method An inhomogeneous DE is a DE of the type 0 + () × = ()
(C.1)
In order to solve Eq. C.1, we first find a solution of the homogeneous DE 0 + () × = 0
(C.2)
and then determine a particular solution of the inhomogeneous DE: • Solution of the homogeneous DE by separation of variables: 0 + () × = 0
(C.3)
0 = − () ×
(C.4)
ln = − () y
= × −
()
(C.5) (C.6)
• Determination of a particular solution of the inhomogeneoues DE by variation of constant: → () (C.7) 0 () =
(C.8)
Insertion of () and 0 () into Eq. C.1 and integration gives =
(C.9)
The general solution of Eq. C.1 is then given by = +
(C.10)
Appendix C
271
C.2 Application to two consecutive first-order reactions Consider the reaction system
(C.11)
(C.12)
1 B A −→
2 C B −→
which is described by the following rate equations [A] = −1 [A] [B] = +1 [A] − 2 [B] [C] = +2 [B]
(C.14)
[A] = [A]0 −1
(C.16)
(C.13)
(C.15)
The solution for Eq. C.13 is
Thus we have to find the solution of the inhomogeneous DE for [B] (Eq. C.14) [B] + 2 [B] = 1 [A]0 −1
(C.17)
• Solution of the homogeneous DE by separation of variables:
y
[B] = −2 [B]
(C.18)
[B] = × −2
(C.19)
• Determination of a particular solution by variation of constant:
y
= ()
(C.20)
[B] = () × −2
(C.21)
[B] () = × −2 − () × 2 −2 Insertion of [B] and
(C.22)
[B] into Eq. C.17 gives
() × −2 − () × 2 −2 + () × 2 −2 = 1 [A]0 −1
(C.23)
which simplifies to () = 1 [A]0 (2 −1 ) This DE is easily integrated to obtain ():
(C.24)
Appendix C
272
— If 1 6= 2 we obtain () = and thus
1 [A]0 (2 −1 ) 2 − 1
[B] = () −2 1 [A]0 (2 −1 ) −2 = 2 − 1 1 [A]0 −1 = 2 − 1
(C.25)
(C.26) (C.27) (C.28)
— If 1 = 2 we obtain () = 1 [A]0 (2 −1 ) = 1 [A]0
(C.29) (C.30)
y () = 1 [A]0
(C.31)
[B] = () −2 = 1 [A]0 −2
(C.32) (C.33)
[B] = [B] + [B]
(C.34)
y
• General solution: y
⎧ 1 [A]0 −1 ⎪ ⎨ × −2 + if 1 = 6 2 2 − 1 [B] = ⎪ ⎩ × −2 + 1 [A]0 −2 if 1 = 2
(C.35)
Initial value condition at = 0:
[B ( = 0)] = 0 y
⎧ [A] ⎪ ⎨ − 1 0 if 1 = 6 2 2 − 1 = ⎪ ⎩ 0 if 1 = 2
(C.36)
(C.37)
Final solutions for [B ()]:
⎧ ¢ 1 [A]0 ¡ −1 ⎪ ⎨ − −2 if 1 = 6 2 [B ()] = 2 − 1 ⎪ ⎩ 1 [A]0 −2 if 1 = 2
(C.38)
Appendix C
273
C.3 Application to parallel and consecutive first-order reactions Consider the reaction system
1 P A −→
(C.39)
1 2 B −→ P A −→
(C.40)
which is described by the following rate equations [A] = − (1 + 1 ) [A] = −1 [A] [B] = +1 [A] − 2 [B] [P] = +1 [A] + 2 [B]
(C.41) (C.42) (C.43)
with 1 = (1 + 1 ).56 The solution for Eq. C.41 is [A] = [A]0 −(1 +1 ) = [A]0 −1
(C.44)
Thus we have to find the solution of the inhomogeneous DE for [B] (Eq. C.42) [B] + 2 [B] = +1 [A]0 −1
(C.45)
using the above standard procedure: • Solution of the homogeneous DE by separation of variables:
y
[B] = −2 [B]
(C.46)
[B] = × −2
(C.47)
• Determination of a particular solution by variation of constant:
y
= ()
(C.48)
[B] = () × −2
(C.49)
[B] () = × −2 − () × 2 −2
56
(C.50)
An example is the radiationless deactivation of a ∗ electronically excited molecule A directly to the ground state P or via an intermediatate optically dark ∗ state B, which decays to the ground state more slowly.
Appendix C
274
Insertion of [B] and
[B] into Eq. C.45 gives
() × −2 − () 2 −2 + 2 () −2 = 1 [A]0 −1
(C.51)
The terms with ± () 2 −2 cancel, so that () × −2 = 1 [A]0 −1
(C.52)
which we rewrite in order to solve () as
or
() = 1 [A]0 −1 × +2
(C.53)
() = 1 [A]0 (2 −1 )
(C.54)
This DE is easily integrated to obtain (): — If 2 6= 1 = (1 + 1 ) we obtain () =
1 [A]0 (2 −1 ) 2 − 1
(C.55)
and thus [B] = () −2 1 [A]0 (2 −1 ) −2 = 2 − 1 1 [A]0 −1 = 2 − 1
(C.56) (C.57) (C.58)
— The case 2 = 1 does not interest us here, as the ∗ state B should be a longer-lived one. • General solution for 2 6= 1 : [B] = [B] + [B] y [B] = × −2 +
1 [A]0 −1 2 − 1
(C.59)
(C.60)
Initial value condition at = 0: [B ( = 0)] = 0
(C.61)
1 [A]0 2 − 1
(C.62)
y =−
Appendix D
275
Final solutions for [B ()]: 1 [A]0 −1 1 [A]0 × −2 + 2 − 1 2 − 1 1 [A]0 −1 1 [A]0 − × −2 = 2 − 1 2 − 1
[B ()] = −
i.e. [B ()] =
¢ 1 [A]0 ¡ −2 × − −1 1 − 2
(C.63) (C.64)
(C.65)
• In the time-resolved fluorescence measurements, we observe () ∝ [A ()] + [B ()]
(C.66)
with some unknown factor accounting for the (lower) transition probability of state B, i.e., the fluorescence-time profile () is ¢ 1 [A]0 ¡ −2 × − −1 1 − 2 1 [A]0 1 [A]0 = [A]0 −1 + × −2 − × −1 1 − 2 1 − 2
() = [A]0 −1 +
¶ µ 1 [A]0 1 [A]0 −2 = × −1 × + [A]0 − 1 − 2 1 − 2 −2
= 2 ×
−1
+ 1 ×
(C.67)
(C.68)
(C.69) (C.70)
This is the same result as would be obtained by a fit to a sum of two exponentials with the rate constants 1 = 1 + 1 and 2 , i.e., time constants 1 = (1 + 1 )−1 and 2 = 2 .
Appendix D
276
Appendix D: Matrix methods This appendix has been taken from the “Introduction to Molecular Spectroscopy” script, not all subsections apply to chemical kinetics.
D.1 Definition I
Matrices: A matrix is a rectangular, × dimensional array of numbers : ⎛
11 12 13 ⎜ 21 22 23 A=⎜ .. .. ⎝ ... . . 1 2 3
The are called the matrix elements.
⎞ 1 2 ⎟ . ⎟ .. ⎠
(D.1)
We will need to consider only × dimensional square matrices, which we can think of as a collection of column vectors a : ⎛ ⎞ 11 12 1 ⎜ 21 22 2 ⎟ (D.2) A=⎜ .. . ⎟ ⎝ ... . .. ⎠ 1 2
I
Notation: Matrices will be denoted by bold symbols: A
I
(D.3)
Matrix manipulation by computer programs: Matrix manipulations are carried out efficiently using computer programs (see, e.g., Press1992) or with symbolic algebra programs.57
57
The most commmon symbolic math programs are MathCad, MuPad, Maple, and Mathematica. MathCad and Mathematica are available in the PC lab.
Appendix D
277
D.2 Special matrices and matrix operations I
Identity matrix: I = 1 ⎛
1 ⎜0 I=1=⎜ ⎝ ... 0
0 1 .. . 0
⎞ 0 0 ⎟ .⎟ .. ⎠ 1
(D.4)
We frequently use the Kronecker : = where
for = for = 6
(D.6)
⎛
⎞ 0 0 ⎟ .. .. ⎟ . . ⎠
(D.7)
⎛
⎞ 0 0 ⎟ .. .. ⎟ . . ⎠
(D.8)
=
I
I
Diagonal matrices:
½
1 0
11 ⎜ 0 22 ⎜ . .. ⎝ .. . 0 0 Block diagonal matrices: 11 12 ⎜ 21 22 ⎜ . .. ⎝ .. . 0 0
I
(D.5)
Complex conjugate (c.c.) of a matrix: A∗ To obtain the complex conjugate A∗ of the matrix A, change to −: (∗ ) = ( )∗
I
(D.9)
Transpose of a matrix: To obtain the transpose A of the matrix A, interchange rows and colums: ¡ ¢ = ( )
(D.10)
Appendix D
I
278
Hermitian conjugate (“transpose conjugate”) of a matrix: A† To obtain the Hermitian conjugate (or “transpose conjugate” or “adjoint”) A† of the matrix A, take the complex conjugate and transpose: ¡ ¢ ¡ ¢∗ † = ∗ = (D.11) where
I
¡ †¢ = ( )∗
Hermitian matrices: A matrix A is Hermitian, if A† = (A∗ ) = A i.e.,
I
(D.12)
¡ †¢ = ( )∗ =
(D.13) (D.14)
Inverse of a matrix: A−1 A matrix A−1 is the inverse of the original matrix A, if AA−1 = A−1 A = I
(D.15)
I
Orthogonal matrices: A matrix A is orthogonal, if the inverse of A equals its transpose: A−1 = A (D.16)
I
Unitary matrices: A matrix U is unitary, if the inverse of U equals its transpose conjugate: U−1 = U† = (U∗ ) (D.17) y
I
U† U = I
Trace of a matrix: tr (A) =
X
(D.18)
(D.19)
=1
I
Matrix addition: C=A+B
(D.20)
= +
(D.21)
with
Appendix D
I
279
Multiplication by a scalar: C = A
(D.22)
=
(D.23)
C = A · B = AB
(D.24)
with
I
Matrix multiplication: with =
X
(D.25)
=1
Example: µ ¶µ ¶ µ ¶µ ¶ 1 2 5 6 1×5+2×7 1×6+2×8 19 22 = 3 4 7 8 3×5+4×7 3×6+4×8 43 50
(D.26)
Notes: (1) Matrix multiplication is defined only if the number of columns of the first matrix is identical to the number of rows of the second matrix. (2) A B 6= B A I
(D.27)
Singular matrices: A matrix A is called singular, if its determinant vanishes: det (A) = |A| = 0
(D.28)
Appendix D
280
D.3 Determinants I
Determinant of a matrix: A determinant of a matrix is, in general, a number which is evaluated according to the rule det (A) = |A| =
X
(D.29)
=1
The are called the co-factors which are obtained by omitting the ’th row and ’th column (marked below in red) and multiplying by (−1)+ : ¯ ¯ ¯ 11 1 1 ¯ ¯ . .. .. ¯¯ ¯ . ¯ . . . ¯ ¯ ¯ (D.30) = (−1)+ ¯ 1 ¯ ¯ . ¯ . . ¯ .. .. .. ¯ ¯ ¯ ¯ 1 ¯ I
Evaluation of simple determinants: ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = −
¯ ¯ ¯ 11 12 13 ¯ ¯ ¯ ¯ 21 22 23 ¯ = ¯ ¯ ¯ 31 32 33 ¯ = 11 22 33 + 12 23 31 + 13 21 32 − (13 22 31 + 11 23 32 + 12 21 33 )
where we “extended” the original determinant according to the scheme ¯ ¯ ¯ 11 12 13 ¯ 11 12 ¯ ¯ ¯ 21 22 23 ¯ 21 22 ¯ ¯ ¯ 31 32 33 ¯ 31 32
(D.31)
(D.32)
(D.33)
Appendix D
281
D.4 Coordinate transformations I
Multiplication of a vector by a matrix: x0 = A x The multiplication of an -dimensional column vector x by an × matrix A gives a new vector x0 with components corresponding to those of x in a transformed coordinate system.
I
Example: Multiplication of a 2-dim vector µ ¶ 1 x= 2 by a matrix A= gives Ax = =
µ
µ
µ
cos − sin sin cos
cos − sin sin cos
¶
¶µ
1 cos − 2 sin 1 sin + 2 cos
= x0
I
(D.34)
Figure D.1: Rotation of a coordinate system.
(D.35)
1 2 ¶
¶
(D.36) (D.37) (D.38)
Appendix D
I
282
The result is a rotation: • In the original coordinate system, the new vector x0 corresponds to the result of an anti-clockwise rotation of x around . • Alternatively, we can say that x0 is given in a new coordinate system that is obtained by a clockwise rotation of the old coordinate system around .
I
Rotation matrices and rotation operators: A is called a rotation matrix. We shall also call A a rotation operator.
I
Exercise D.1: Show that Eqs. D.36 - D.38 does describe an anti-clockwise rotation of the vector x to the new vector x0 and determine its coordinates using trigonometric arguments. ¤
Appendix D
283
D.5 Systems of linear equations I
Solutions of linear equations (I): Cramer’s rule. Matrices and determinants can be generally used for solving systems of linear equations of the type 11 1 + 12 2 + 13 3 + 1 21 1 + 22 2 + 23 3 + 2 1 1 + 2 2 + 3 3 +
= = = =
1 2
These equations can be written in matrix form: ⎞⎛ ⎞ ⎛ ⎞ ⎛ 1 1 11 12 1 ⎜ 21 22 2 ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ . ⎟=⎜ . ⎟ ⎜ . .. .. ⎟ ⎝ .. . . ⎠ ⎝ .. ⎠ ⎝ .. ⎠ 1 2
(D.39)
(D.40)
or
Ax = c
(D.41)
Simple algebraic manipulations (see any math textbook) gives the expressions |A| = |A |
(D.42)
where |A| is the determinant of the coefficient matrix A and |A | is the respective determinant of |A| in which the th column is replaced by the 1 2 3 , e.g. ⎞ ⎛ 11 1 1 ⎜ 21 2 2 ⎟ (D.43) |A2 | = ⎜ .. ⎟ ⎝ ... ... . ⎠ 1
Solutions for the : From Eq. D.42 we obtain the non-trivial solutions for the according to |A | (D.44) = |A| under the condition that the A matrix is not singular, i.e., the determinant of coefficients does not vanish |A| 6= 0 (D.45) (The trivial and uninteresting solutions are 1 2 3 = 0).
I
Solutions of linear equations (II): In the following, we shall only consider the special linear equations of the type 11 1 + 12 2 + 13 3 + 1 = 0 21 1 + 22 2 + 23 3 + 2 = 0 . = .. 1 1 + 2 2 + 3 3 + = 0
(D.46)
Appendix D
284
or Ax = 0
(D.47)
The trivial and uninteresting solutions of this equation are 1 2 3 = 0. The interesting solutions require that the matrix of coefficients A is singular, i.e., the determinant of coefficients has to vanish |A| = 0 This type of equations is encountered in eigenvalue problems.
(D.48)
Appendix D
285
D.6 Eigenvalue equations Consider a set of linear equations as considered in the previous section which can be written in the form A x = x (D.49) where (1) A is an × -dimensional matrix, (2) is a scalar constant, called an eigenvalue of A ( is any one of the set of eigenvalues of A), and (3) x is an -dimensional column vector (which can in general be complex), called the eigenvector of A belonging to the particular eigenvalue. Eq. D.49 is called an eigenvalue equation. It has the following property: The multiplication of x by (and hence that of x by the matrix A) changes the length of x (by the factor ), but not the direction. I
Eigenvalues and eigenvectors of the molecular Hamiltonian: The determination of eigenvalues and eigenvectors of the molecular Hamiltonian H, i.e., the “energy b is the most important problem in matrix”, or the matrix of the Hamilton operator , spectroscopy (in general, it is the key problem!!).
I
Secular equation and eigenvalues λ : Eq. D.49 can be rewritten as (A − I) x = 0
(D.50)
In order not to obtain the trivial solutions 1 = 2 = 3 = = 0, det (A − I) has to vanish, i.e., det (A − I) = 0. This is the so-called characteristic equation or secular equation: ¯ ¯ 11 − 12 ¯ ¯ 21 22 − |A − I| = ¯¯ .. .. . . ¯ ¯ 2 1
The secular equation has roots
¯ 1 ¯ ¯ 2 ¯ ¯=0 .. ¯ . ¯ − ¯
{1 2 } which are called the eigenvalues.
(D.51)
(D.52)
Appendix D
I
286
Eigenvectors x : For each eigenvalue , we can write an eigenvalue equation A x1 = 1 x1 A x2 = 2 x2 .. .
(D.53) (D.54)
A x = x
(D.55) (D.56)
AX=XΛ
(D.57)
In matrix notation, this becomes
with the diagonal eigenvalue matrix ⎛
1 ⎜ 0 Λ=⎜ ⎝ ... 0
0 2 .. . 0
⎞ 0 0 ⎟ . ⎟ .. ⎠
(D.58)
The different eigenvectors x are usually normalized to unity, i.e., ¢12 ¡ =1 |x| = x† x I
(D.59)
Matrix diagonalization: The matrix of the eigenvectors X can be determined by multiplying the equation AX=XΛ (D.60) from the left with X−1 , which gives X−1 A X = X−1 X Λ
(D.61)
X−1 A X = Λ
(D.62)
i.e., y The determination of the eigenvector matrix is equivalent to finding a matrix X that transforms the matrix A into diagonal form (Λ). I
Example: A=
µ
2 3 3 10
¶
(D.63)
Eigenvalue equations: A x = x
(D.64)
AX=XΛ
(D.65)
or in matrix form:
Appendix E
287
Secular equation: ¯ ¯ ¯2− 3 ¯¯ ¯ det (A − I) = ¯ 3 10 − ¯ = (2 − ) (10 − ) − 9 = 0 y
(D.66) (D.67)
0 = 20 − 2 − 10 + 2 − 9 = 2 − 12 + 11 12
12 = + ± 2 = 6±5
sµ
12 2
¶2
− 11 = 6 ±
(D.68)
√ 36 − 11
(D.69) (D.70)
y Eigenvalues: (D.71) (D.72)
1 = 1 2 = 11
Eigenvectors: Inserting the eigenvalues one by one into the eigenvalue equation yields: (1)
y
(2)
y
µ
µ
2 3 3 10
2 3 3 10
¶µ
11 21
¶µ
¶
12 22
¶
= 1 µ
µ
11 21
−3 1
¶
= 2
µ
¶
= 11
µ
11 21
¶
(D.73)
(D.74)
12 22
µ ¶ 1 3
¶
=1
µ
12 22
¶
(D.75)
(D.76)
Unnormalized eigenvalue matrix: µ
−3 1 1 3
¶
(D.77)
Normalized eigenvalue matrix: ⎛ −3 1 ⎞ √ √ ⎟ ⎜ X = ⎝ 110 310 ⎠ √ √ 10 10
(D.78)
Appendix E
288
I
Final note: In practice, matrix manipulations such as matrix inversion, the evaluation of determinants, the determination of eigenvalues and eigenvectors, etc., are carried out numerically by using computers.
I
References Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1992. Versions are also available for C and Pascal.
Appendix E
289
Appendix E: Laplace transforms The concept of Laplace and inverse Laplace transforms is extremely useful in chemical kinetics for two reasons: (1) They connect microscopic molecular properties and statistically averaged quantities: a) () ↔ ( ): Collision cross section vs. thermal rate constant for bimolecular reactions (section ??), b) () ↔ ( ): Specific rate constant vs. thermal rate constant for unimolecular reactions (section 8), c) () ↔ ( ): Density of states vs. partition function in statistical rate theories 8). (2) They provide a convenient method for solving of differential equations (section 3.4.2). I
Definition E.1: The Laplace transform L [ ()] of a function () is defined as the integral Z∞ () = L [ ()] = () − (E.1) 0
where • is a real variable, • () is a real function of the variable with the property () = 0 for 0, • is a complex variable, • () = L [ ()] is a function of the variable .
I
Definition E.2: The inverse Laplace transform L−1 [ ()] of the function () is defined as the integral −1
() = L
1 [ ()] = 2
+∞ Z
−∞
where • is an arbitrary real constant.
()
(E.2)
Appendix F
I
290
Notes: • Since L−1 [ ()] recovers the original function (), the pair of functions () and () is said to form a Laplace pair. • The properties of the Laplace and inverse Laplace transforms can be derived from those of Fourier transforms (Zachmann 1972). • Laplace transforms can be computed using symbolic algebra computer programs (Maple, Mathematica, etc.). In favorable cases, it is also possible to obtain inverse Laplace transforms by computer programs. However, it is often more convenient to take the Laplace and inverse Laplace transforms from corresponding Tables.58
I
Table E.1: Table of Laplace transforms. ()
() = L [ ()]
() () =
R∞ 0
()
() = L [ ()]
1 −
() −
()
() − ( = 0)
1
1
1 2
sin
2 2
1 3
cos
−1 Γ ()
1
1 ( − )2 2
+ 2
2
+ 2
¢ 1 ¡ 1 − ( − ) ( − ) ( − )
Γ () is the Gamma function. For integer , Γ () = ( − 1)! (see Appendix F). I
References: Houston 1996 P. L. Houston, Chemical Kinetics and Reaction Dynamics, McGrawHill, Boston, 2001. Steinfeld 1989 J. I. Steinfeld, J. S. Francisco, W. L. Haase, Chemical Kinetics and Dynamics, Prentice Hall, Englewood Cliffs, 1989. Zachmann 1972 H. G. Zachmann, Mathematik für Chemiker, Verlag Chemie, Weinheim, 1972.
58
Extensive Tables of Laplace transforms are given by (Steinfeld 1989).
Appendix F
291
Appendix F: The Gamma function Γ (x) • If is a positive integer, the Gamma function is Γ () = ( − 1)!
(F.1)
• The Gamma function smoothly interpolates between all points ( ) given by = ( − 1)! for positive non-integer (cf. Fig. F.1). • The Gamma function is defined for all complex numbers with a positive real part as Z ∞ −1 − (F.2) Γ () = 0
This integral converges for complex numbers with a positive real part. It has poles for complex numbers with a negative real part (cf. Fig. F.2).
I
Figure F.1: The Gamma function.
Appendix G
I
292
Figure F.2: For negative arguments the Gamma function has poles (at all negative integers).
Appendix G
293
Appendix G: Ergebnisse der statistischen Thermodynamik In diesem Kapitel sollen wichtige Ergebnisse der statistischen Thermodynamik zusammengefasst werden. Im Vordergrund steht die Berechnung der thermodynamischen Zustandsfunktionen und die Berechnung von chemischen Gleichgewichten aus molekularen Eigenschaften. Wir benutzen die Ergebnisse der Quantenmechanik: • Ein Molekül kann nur bestimmte Energiezustände einnehmen. • Diese Energiezustände sind im Prinzip nicht kontinuierlich sondern diskret verteilt. • Jeder Energiezustand hat ein bestimmtes statistisches Gewicht . Das statistische Gewicht gibt an, wie oft ein Zustand bei einer bestimmten Energie vorkommt. Beispiele: eindimensionaler harmonischer Oszillator isotroper zweidimensionaler harmonischer Oszillator (z.B. Biegeschwingung von CO2 ) = 2 + 1 Rotation eines linearen Moleküls
= 1 = 2
(G.1)
• Die Ergebnisse der statischen Thermodynamik unterscheiden sich, je nachdem ob das betrachtete System aus unterscheidbaren oder aus ununterscheidbaren Teilchen besteht: Beispiele für unterscheidbare Teilchen: Atome im Kristallgitter Beispiele für ununterscheidbare Teilchen: 1.) Gasmoleküle 2.) Elektronen (Fermi-Statistik) (G.2) • Die anzuwendende Statistik hängt vom Spin der Teilchen ab: Spin gerade: Bose-Einstein-Statistik Spin ungerade: Fermi-Dirac-Statistik
(G.3)
Auf die Unterschiede zwischen der Bose-Einstein- und der Fermi-Dirac-Statistik wird an anderer Stelle eingegangen (⇒ Vorlesung “Statistische Thermodynamik”). • Im Grenzfall À gilt die Boltzmann-Statistik. • Im Folgenden werden behandelt: Elektronische Anregung, Schwingungsanregung, Rotationsanregung, Translationsanregung. Die Translation erfordert abhängig vom Problem manchmal eine etwas andere Behandlung (klassische statistische Thermodynamik bzw. Semiklassik). Quantenmechanik und klassische Mechanik stimmen im Ergebnis für À überein.
Appendix G
294
G.1 Boltzmann-Verteilung Die Besetzungswahrscheinlichkeit eines Zustands wird durch die Boltzmann-Verteilung angegeben. Diese ergibt sich als die wahrscheinlichste Verteilung aus dem BoltzmannAnsatz für die Entropie = ln (G.4) wird als die thermodynamische Wahrscheinlichkeit bezeichnet. Die Boltzmann-Verteilung kann hergeleitet werden, wenn man entsprechend dem 2. Hauptsatz den Maximalwert für sucht (Methode der Lagrange’schen Multiplikatoren). Hier setzen wir die Boltzmann-Verteilung als gegeben voraus (z.B. als empirisches, experimentelles Ergebnis): I
Boltzmann-Verteilung: Für diskrete Energiezustände zungswahrscheinlichkeit eines Zustands proportional zu − : =
∝ −
ist
die
Beset-
(G.5)
Bei Berücksichtigung einer möglichen Entartung von Zuständen (Entartungsfaktor bzw. statistisches Gewicht ) und mit einer Proportionalitätskonstante 0 zur Normierung gilt = 0 × × − (G.6) = Dabei ist die Zahl der Teilchen im Energiezustand (mit dem statistischen Gewicht ) und die Gesamtzahl der Teilchen. Die Funktion gibt die Wahrscheinlichkeit der Besetzung des Zustands an. Die Normierungskonstante 0 ergibt sich aus der Normierungsbedingung, dass die Gesamtwahrscheinlichkeit zu 1 herauskommt: P X X = = 0 × − = 1 (G.7) 1 . −
y I
y 0 = P
Normierte Boltzmann-Verteilungsfunktion: =
I
(G.8)
− =P −
(G.9)
P Molekulare Zustandssumme: Die Summe − im Nenner der o.a. Gleichung wird als molekulare Zustandssumme bezeichnet: =
P
−
(G.10)
Die Zustandssumme ist die zentrale Grösse der statistischen Thermodynamik. ist von der Temperatur abhängig.
Appendix G
I
295
Anschauliche Bedeutung der molekularen Zustandssumme: = Zahl der bei der Temperatur im Mittel besetzten Energiezustände eines Molekül.
I
Boltzmann-Verteilungsfunktion für kontinuierlich verteilte Energiezustände: Für kontinuierlich verteilte Energiezustände ersetzt man durch () und erhält analog eine kontinuierliche Besetzungswahrscheinlichkeit: () () × − () = =
I
(G.11)
Zustandsdichte: () = Zustandsdichte (=Anzahl der Zustände pro EnergieIntervall): () () = (G.12)
Appendix G
296
G.2 Mittelwerte I
Mittlere Energie ²: In der Thermodynamik interessieren Mittelwerte, z.B. die mittlere Energie . Für Größen, die durch eine Verteilungsfunktion beschrieben werden, erhält man den Mittelwert nach P P P P X = P = P = (G.13) = = P
mit =
P und = 1.
Übergang zu einer kontinuierlichen Verteilungsfunktion liefert Z∞ ()
= 0Z∞ ()
(G.14)
0
Diese Ausdrücke vereinfachen sich, wenn () auf 1 normiert ist. I
Allgemeine Berechnung der Mittelwerts y einer Größe y(²): X = X
(G.15)
bzw.
=
Z∞ ()() 0
Z∞ ()
(G.16)
0
I
Example G.1: Berechnung der mittleren Energie für ein 3-Niveausystem: 0 = 0 1 = 1 2 = 21 0 = 3 2 = 2 2 = 1 =
X
= 6
(G.17)
(G.18)
Appendix G
297
Ergebnis: = = = = = =
X X 1 X = = 1 (30 + 21 + 12 ) 6 1 (30 + 21 + 21 ) 6 1 (0 + 41 ) 6 4 1 6 2 1 3
(G.19) (G.20) (G.21) (G.22) (G.23) (G.24) ¤
I
Mittlere Energie eines Moleküls: Bei Gültigkeit der Boltzmann-Verteilung kann die mittlere Energie eines Moleküle aus der molekularen Zustandssumme berechnet werden: = = = = = = =
X X 1 X = = (G.25) P − P − (G.26) P 2 − |× (G.27) 2 − 2 X (G.28) 2 2 X ¡ − ¢ |(durch differenzieren zeigen) (G.29) 2 X − (G.30) 2
(G.31)
y = 2
ln
(G.32)
Appendix G
298
G.3 Anwendung auf ein Zweiniveau-System I
Chemisch relevante Beispiele für Zweiniveau-Systeme: • Paramagnetische Atome u. Moleküle • NMR-Spektroskopie: 100 MHz.
Atomkerne mit Kernspin.
Typischer Frequenzbereich:
• Molekül mit cis-trans-Isomeren oder 2 Konformeren
I
Figure G.1: Zwei-Niveau-System: Energiezustände The Two Level System: Energy Levels
Zustandssumme:
g
Q gi e i / kT i
1 e 0 / kT 1 e 1 / kT 1 e 1 / kT Besetzungsverhältnis: N 0 g0e 0 / kT N Q N1 g1e 1 / kT N Q
g
N1 g1e 1 / kT e 1 / kT N 0 g0e 0 / kT
Physical Chemistry III: Chemical Kinetics • © F. Temps, IPC Kiel
I
173
Besetzungsverhältnis: 1 −1 1 =
(G.33)
= 0 −0 + 1 −1 = 1 + −1
(G.34) (G.35)
1 → ∞ y −1 → −∞ = 0
(G.36)
mit
(1) Grenzfall für → 0:
y Bei → 0 ist nur der tiefste Energiezustand besetzt.
Appendix G
299
(2) Grenzfall für → ∞:
1 (G.37) → 0 y −1 → −0 = 1 y Bei → ∞ ergibt sich eine Gleichbesetzung der Energiezustände. Dies ist die maximale Besetzung im oberen Zustand! • Eine Besetzungsinversion (1 0 ) würde formal eine negative Temperatur erfordern (→ Laser).
I
Figure G.2: Zwei-Niveau-System: Boltzmann-Besetzung der Zustände (rot: unterer Zustand, blau: oberer Zustand). The Two Level System: Boltzmann Distribution
Ni N
= 1000 J/mol
1
0.75
0.5
0.25
0 0
250
500
750
1000
T / (K)
Physical Chemistry III: Chemical Kinetics • © F. Temps, IPC Kiel
I
174
Mittlere thermische Energie: (1) 1 = 125 kJ mol (entsprechend = 1000 cm−1 ; z.B. C-C-Schwingung im IR oder Energieabstand zwischen cis-trans-Isomeren): ½ 1 8 × 10−3 @ = 300 K −1 (G.38) = = 023 @ = 1000 K 0 (2) 1 = 004 kJ mol (entsprechend Energieabstand zwischen Kernspinzuständen bei 100 MHz NMR): 1 = −1 = 0999984 @ = 300 K 0
(G.39)
Reihenentwicklung: 1 1 + ≈ 1 − (G.40) y sehr kleiner Besetzungsunterschied y Wunsch der NMR-Spektroskopiker zu immer größeren Frequenzen (Rekord heute 950 MHz). −1 = 1 −
Appendix G
I
300
Figure G.3: Zwei-Niveau-System: Mittlere thermische Energie. The Two Level System: Energy
E
= 1000 J/mol
1000
750
500
250
0 0
250
500
750
1000
T / (K)
Physical Chemistry III: Chemical Kinetics • © F. Temps, IPC Kiel
I
175
Spezifische Wärme: =
I
(G.41)
Figure G.4: Zwei-Niveau-System: Spezifische Wärme. The Two Level System: Specific Heat
cv
= 1000 J/mol
5
4
3
2
1
0 0
250
500
750
1000
T / (K)
Physical Chemistry III: Chemical Kinetics • © F. Temps, IPC Kiel
176
Appendix G
301
G.4 Mikrokanonische und makrokanonische Ensembles Die statistische Thermodynamik macht Aussagen über statistische Ensembles (Ansammlungen von Systemen, Molekülen). Wir betrachten verschiedene Ensembles von miteinander wechselwirkenden Molekülen. I
Der Begriff des Ensembles: Wir betrachten ein abgeschlossenes System bei konstantem Volumen, konstanter Zusammensetzung und konstanter Temperatur (bzw. bestimmter Energie (s.u.)). Ein Ensemble erhalten wir, wenn wir uns N Replikationen dieser Systeme denken, die wir unter Einhaltung bestimmter Regeln erhalten (Regel = Canon). Alle Systeme in dem Ensemble sollen im thermischen Gleichgewicht miteinander (sie dürfen zwischen einander Energie austauschen). Es werden verschiedene Sorten von Ensembles unterschieden: • Mikrokanonisches Ensemble: Ensemble von Systemen (z.B. Atomen, Molekülen) mit konstanter Teilchenzahl, konstantem Volumen, konstanter Energie. • Makrokanonisches Ensemble (kurz kanonisches Ensemble genannt): Ensemble von Systemen (z.B. Atomen, Molekülen) mit konstanter Teilchenzahl, konstantem Volumen, konstanter Temperatur. • Großkanonisches Ensemble: Ensemble von offenen Systemen (z.B. Teilchen, Molekülen) mit konstantem Volumen, konstanter Temperatur, die zwischeneinander Teilchen austauschen können.
I
Systemzustandssumme (= kanonische Zustandssumme): Molekulare Zustandssumme: P = alle Molekülzustände − (G.42)
Systemzustandssumme (= kanonische Zustandssumme): =
P
alle Systemzustände
−
(G.43)
Innere Energie des Systems: = 2
ln
(G.44)
Bei mehreren Teilchensorten (A, B, C, . . . ): Zustandsummen sind multiplikativ, da die Energien additiv sind, y = × × ×
(G.45)
Appendix G
302
• Systemzustandssumme für unterscheidbare Teilchen der Sorte (z.B. Kristall): Y = = (G.46)
y für 1 mol: = 2
ln ln = 2
(G.47)
• Systemzustandssumme für nicht unterscheidbare Teilchen der Sorte (z.B. Gas): = !
(G.48)
y für 1 mol ( = ): = 2
ln = 2 ln !
(G.49)
Appendix G
303
G.5 Statistische Interpretation der thermodynamischen Zustandsgrößen I
Innere Energie: = 2
I
ln
(G.50)
Helmholtz-Energie: = −
µ ¶
(G.51)
¶ − µ ¶ = − 2 − ln = = − − = − 2 2 µ
y
(G.52) (G.53)
= − ln
(G.54)
= − ln
(G.55)
y
I
Entropie:
µ
=− I
Druck:
µ
=−
¶
¶
=
=
− =
µ
ln
¶
=
(G.56)
(G.57)
Appendix G
304
G.6 Chemisches Gleichgewicht
y bzw.
I
+ À
(G.58)
µ ¶ ∆ ª = exp −
(G.59)
∆ ª = − ln
(G.60)
Gleichgewichtskonstante: Wir berechnen zunächst = − ln ,
(G.61)
indem wir = etc.. . . einsetzen und die Stirling’sche Näherung ln ! = ! ln − anwenden:
= − [ln + ln + ln ] " # = − ln + ln + ln ! ! ! = − [( ln − ln !) + ( ln − ln !) + ( ln − ln !)] = − [( ln − ln + ) + ( ln − ln + ) + ( ln − ln + )] I
Gleichgewichtsbedingung: Im chemischen Gleichgewicht muss gelten µ ¶ =0
(G.62) (G.63)
(G.64)
(G.65)
(G.66)
y [( ln − ln + ) + ( ln − ln + ) + ( ln − ln + )] µ ¶ 1 = ln − − ln + 1 ¶ µ 1 − ln + 1 + ln − ¶ µ 1 − ln + 1 − ln − = (ln − ln ) + (ln − ln ) − (ln − ln ) − ln = ln
0 =
(G.67)
(G.68) (G.69) (G.70)
Appendix G
305
y
=
(G.71)
Übergang auf Teilchendichten und volumenbezogenen Zustandssummen: ( ) ( ) = ( ) ( ) ( ) ( ) y =
( ) ∗ = ∗ ∗ ( ) ( )
(G.72)
(G.73)
Bezug auf die getrennten molekularen Nullpunktsenergien: 0 = 0 0 × −∆0
(G.74)
mit 0 = die auf das Volumen bezogene Molekülzustandssumme: 0 =
I
× × ×
(G.75)
Endergebnis: Die Indices ∗ und 0 wurden hier zur Klarheit benutzt. Der Einfachheit halber werden diese üblicherweise weggelassen. y =
× −∆0
(G.76)
Appendix G
306
G.7 Zustandssumme für elektronische Zustände Die elektronische Zustandssumme muss unter Einschluss aller elektronischen Entartungen explizit berechnet werden: =
X
−
(G.77)
= 2 + 1
(G.78)
Speziell sind zu berücksichtigen: • Spinentartung (Spinmultiplizität ):
• elektronischer Bahndrehimpuls ( (Atome) bzw. Λ (lin. Moleküle) bzw. Symmetrierasse oder für nichtlin. Moleküle): • elektronische Feinstrukturkomponenten (Index im Atom-Termsymbol): = 2 + 1
(G.79)
Nur niedrig liegende Elektronenzustände liefern einen Beitrag. Die Zustandsumme ist explizit auszurechnen.
Appendix G
307
G.8 Zustandssumme für die Schwingungsbewegung I
Harmonischer Oszillator: (G.80)
= = 0 1 2 I
Schwingungszustandssumme: =
∞ X
− =
∞ X
−· mit =
=0 ©=0 − ª −2 = 1+ + + −3 + mit = − 1 2
y
= 1 + + + + 1 für 1 = 1−
I
(G.83) (G.84)
(G.85)
µ ¶¸−1 ∙ Y 1 − exp − = =1
(G.86)
s Oszillatoren:
typische Werte bei Zimmertemperatur: ≈ 1 10
I
(G.82)
¶¸−1 ∙ µ 1 = = 1 − exp − 1 − −
I
3
(G.81)
(G.87)
Grenzwert Q für T → ∞ bzw. für ν → 0: Für → ∞ bzw. für → 0 µ von ¶ kann exp − entwickelt werden: µ ¶ exp − ≈1− + (G.88) y = y
1 1 µ µ ¶= ¶ 1 − exp − 1− 1− + ≈
(G.89)
(G.90)
Für klassische Oszillatoren: ≈
Y =1
(G.91)
Appendix G
I
308
Abweichungen vom harmonischen Oszillator: • Für anharmonische Oszillatoren am besten durch explizite Pist die−Zustandssumme zu ermitteln. numerische Berechnung nach ∞ =0
• Torsionsschwingungen können bei tiefen Temperaturen als harmonische Oszillatoren beschrieben werden, bei hohen Temperaturen als freie 1D-Rotatoren. Im Zwischenbereich muss die Zustandssumme ebenfalls explizit berechnet werden (wobei die Energiezustände für den gehinderten 1D-Rotator zugrundegelegt werden müssen).
Appendix G
309
G.9 Zustandssumme für die Rotationsbewegung I
lineare Moleküle: =
I
~2 ( + 1) = 0 1 2 2 ~2 = 2
(G.92) (G.93)
Rotationszustandssumme: 1X (2) = − =0 ∞
1X 2 (2 + 1) −~ (+1)2 =0 Z∞ 1 2 (2 + 1) −~ (+1)2 = ∞
=
(G.94) (G.95)
(G.96)
=0
y (2) =
2 = ~2
(G.97)
(G.98)
I
Symmetriezahl:
I
Rotationszustandssumme für nichtlineare Moleküle: ist nur für symmetrische Kreisel analytisch darstellbar, nicht für asymmetrische Kreisel. Für nicht zu tiefe Temperaturen ist 12 (3) =
I
µ
2 ~2
¶32
( )12 =
12 ( )32 ( )−12
(G.99)
typische Werte bei Zimmertemperatur: ≈ 100 1000
(G.100)
Appendix G
310
G.10 Zustandssumme für die Translationsbewegung I
Teilchen im 1D-Kasten: ³ ´2 2 × = 1 2 3 8
=
I
1D-Translationszustandssumme: =
∞ X
−
=
µ
2 2
y =
I
2 × = 8
"µ
µ
(G.102)
1
¡ ¢ in cm−1
×
2 2
¶12
¶2
+
µ
¶2
3DTranslationszustandssumme: ¶32 µ 2 × = 2 y =
I
¶12
Z∞ 2 2 2 = − 8
(G.103)
(G.104)
Teilchen im 3D-Kasten:
I
(G.101)
µ
2 2
+
µ
¶2 #
¡ ¢ in cm−3 ¶32
(G.105)
(G.106)
(G.107)
typische Werte: y
≈ 1024 ×
(G.108)
≈ 1024 cm−3
(G.109)
